[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
[PROOFSTEP]
have lnonneg : 0 ≤ l := by
  rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
  have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) := tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)
  apply le_of_tendsto_of_tendsto' this clim fun n => _
  simp_rw [div_eq_inv_mul]
  exact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ 0 ≤ l
[PROOFSTEP]
rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
c : ℕ → ℕ
left✝ : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ 2 * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ 0 ≤ l
[PROOFSTEP]
have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) := tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
c : ℕ → ℕ
left✝ : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ 2 * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
this : Tendsto (fun n => u 0 / ↑(c n)) atTop (𝓝 0)
⊢ 0 ≤ l
[PROOFSTEP]
apply le_of_tendsto_of_tendsto' this clim fun n => _
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
c : ℕ → ℕ
left✝ : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ 2 * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
this : Tendsto (fun n => u 0 / ↑(c n)) atTop (𝓝 0)
⊢ ∀ (n : ℕ), u 0 / ↑(c n) ≤ u (c n) / ↑(c n)
[PROOFSTEP]
simp_rw [div_eq_inv_mul]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
c : ℕ → ℕ
left✝ : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ 2 * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
this : Tendsto (fun n => u 0 / ↑(c n)) atTop (𝓝 0)
⊢ ∀ (n : ℕ), (↑(c n))⁻¹ * u 0 ≤ (↑(c n))⁻¹ * u (c n)
[PROOFSTEP]
exact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
[PROOFSTEP]
have A : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, u n - n * l ≤ ε * (1 + ε + l) * n :=
  by
  intro ε εpos
  rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
  have L : ∀ᶠ n in atTop, u (c n) - c n * l ≤ ε * c n :=
    by
    rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
    filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
    have cnpos : 0 < c n := cnpos'
    calc
      u (c n) - c n * l = (u (c n) / c n - l) * c n := by
        simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
      _ ≤ ε * c n := by
        refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
        simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
        exact le_trans (le_abs_self _) hn
  obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
    eventually_atTop.1 (cgrowth.and L)
  let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
  filter_upwards [Ici_mem_atTop M] with n hn
  have exN : ∃ N, n < c N := by
    rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
    exact ⟨N, by linarith only [hN]⟩
  let N := Nat.find exN
  have ncN : n < c N := Nat.find_spec exN
  have aN : a + 1 ≤ N := by
    by_contra' h
    have cNM : c N ≤ M := by
      apply le_max'
      apply mem_image_of_mem
      exact mem_range.2 h
    exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
  have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
  have cNn : c (N - 1) ≤ n := by
    have : N - 1 < N := Nat.pred_lt Npos.ne'
    simpa only [not_lt] using Nat.find_min exN this
  have IcN : (c N : ℝ) ≤ (1 + ε) * c (N - 1) :=
    by
    have A : a ≤ N - 1 := by
      apply @Nat.le_of_add_le_add_right a 1 (N - 1)
      rw [Nat.sub_add_cancel Npos]
      exact aN
    have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
    have := (ha _ A).1
    rwa [B] at this
  calc
    u n - n * l ≤ u (c N) - c (N - 1) * l := by
      apply sub_le_sub (hmono ncN.le)
      apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
    _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
    _ ≤ ε * c N + ε * c (N - 1) * l := by
      apply add_le_add
      · apply (ha _ _).2
        exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
      · apply mul_le_mul_of_nonneg_right _ lnonneg
        linarith only [IcN]
    _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l := (add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
    _ = ε * (1 + ε + l) * c (N - 1) := by ring
    _ ≤ ε * (1 + ε + l) * n := by
      refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
      apply mul_nonneg εpos.le
      linarith only [εpos, lnonneg]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
intro ε εpos
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have L : ∀ᶠ n in atTop, u (c n) - c n * l ≤ ε * c n :=
  by
  rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
  filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
  have cnpos : 0 < c n := cnpos'
  calc
    u (c n) - c n * l = (u (c n) / c n - l) * c n := by
      simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
    _ ≤ ε * c n := by
      refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
      simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
      exact le_trans (le_abs_self _) hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
[PROOFSTEP]
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
⊢ ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
[PROOFSTEP]
filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
⊢ u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
[PROOFSTEP]
have cnpos : 0 < c n := cnpos'
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
[PROOFSTEP]
calc
  u (c n) - c n * l = (u (c n) / c n - l) * c n := by
    simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
  _ ≤ ε * c n := by
    refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
    simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
    exact le_trans (le_abs_self _) hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ u (c n) - ↑(c n) * l = (u (c n) / ↑(c n) - l) * ↑(c n)
[PROOFSTEP]
simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ (u (c n) / ↑(c n) - l) * ↑(c n) ≤ ε * ↑(c n)
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ u (c n) / ↑(c n) - l ≤ ε
[PROOFSTEP]
simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
hn : |u (c n) / ↑(c n) - l| ≤ ε
⊢ u (c n) / ↑(c n) - l ≤ ε
[PROOFSTEP]
exact le_trans (le_abs_self _) hn
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
⊢ ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
  eventually_atTop.1 (cgrowth.and L)
[GOAL]
case intro.intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
⊢ ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
⊢ Finset.Nonempty (image (fun i => c i) (range (a + 1)))
[PROOFSTEP]
simp
[GOAL]
case intro.intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
⊢ ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
filter_upwards [Ici_mem_atTop M] with n hn
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have exN : ∃ N, n < c N := by
  rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
  exact ⟨N, by linarith only [hN]⟩
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
⊢ ∃ N, n < c N
[PROOFSTEP]
rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
N : ℕ
hN : n + 1 ≤ c N
⊢ ∃ N, n < c N
[PROOFSTEP]
exact ⟨N, by linarith only [hN]⟩
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
N : ℕ
hN : n + 1 ≤ c N
⊢ n < c N
[PROOFSTEP]
linarith only [hN]
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
let N := Nat.find exN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have ncN : n < c N := Nat.find_spec exN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have aN : a + 1 ≤ N := by
  by_contra' h
  have cNM : c N ≤ M := by
    apply le_max'
    apply mem_image_of_mem
    exact mem_range.2 h
  exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
⊢ a + 1 ≤ N
[PROOFSTEP]
by_contra' h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ False
[PROOFSTEP]
have cNM : c N ≤ M := by
  apply le_max'
  apply mem_image_of_mem
  exact mem_range.2 h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ c N ≤ M
[PROOFSTEP]
apply le_max'
[GOAL]
case H2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ c N ∈ image (fun i => c i) (range (a + 1))
[PROOFSTEP]
apply mem_image_of_mem
[GOAL]
case H2.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ N ∈ range (a + 1)
[PROOFSTEP]
exact mem_range.2 h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
cNM : c N ≤ M
⊢ False
[PROOFSTEP]
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have cNn : c (N - 1) ≤ n := by
  have : N - 1 < N := Nat.pred_lt Npos.ne'
  simpa only [not_lt] using Nat.find_min exN this
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ c (N - 1) ≤ n
[PROOFSTEP]
have : N - 1 < N := Nat.pred_lt Npos.ne'
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
this : N - 1 < N
⊢ c (N - 1) ≤ n
[PROOFSTEP]
simpa only [not_lt] using Nat.find_min exN this
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
have IcN : (c N : ℝ) ≤ (1 + ε) * c (N - 1) :=
  by
  have A : a ≤ N - 1 := by
    apply @Nat.le_of_add_le_add_right a 1 (N - 1)
    rw [Nat.sub_add_cancel Npos]
    exact aN
  have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
  have := (ha _ A).1
  rwa [B] at this
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
have A : a ≤ N - 1 := by
  apply @Nat.le_of_add_le_add_right a 1 (N - 1)
  rw [Nat.sub_add_cancel Npos]
  exact aN
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
⊢ a ≤ N - 1
[PROOFSTEP]
apply @Nat.le_of_add_le_add_right a 1 (N - 1)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
⊢ a + 1 ≤ N - 1 + 1
[PROOFSTEP]
rw [Nat.sub_add_cancel Npos]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
⊢ a + 1 ≤ N
[PROOFSTEP]
exact aN
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
A : a ≤ N - 1
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
A : a ≤ N - 1
B : N - 1 + 1 = N
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
have := (ha _ A).1
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
A : a ≤ N - 1
B : N - 1 + 1 = N
this : ↑(c (N - 1 + 1)) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
rwa [B] at this
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
calc
  u n - n * l ≤ u (c N) - c (N - 1) * l := by
    apply sub_le_sub (hmono ncN.le)
    apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
  _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
  _ ≤ ε * c N + ε * c (N - 1) * l := by
    apply add_le_add
    · apply (ha _ _).2
      exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
    · apply mul_le_mul_of_nonneg_right _ lnonneg
      linarith only [IcN]
  _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l := (add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
  _ = ε * (1 + ε + l) * c (N - 1) := by ring
  _ ≤ ε * (1 + ε + l) * n := by
    refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
    apply mul_nonneg εpos.le
    linarith only [εpos, lnonneg]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ u n - ↑n * l ≤ u (c N) - ↑(c (N - 1)) * l
[PROOFSTEP]
apply sub_le_sub (hmono ncN.le)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ↑(c (N - 1)) * l ≤ ↑n * l
[PROOFSTEP]
apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ u (c N) - ↑(c (N - 1)) * l = u (c N) - ↑(c N) * l + (↑(c N) - ↑(c (N - 1))) * l
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ u (c N) - ↑(c N) * l + (↑(c N) - ↑(c (N - 1))) * l ≤ ε * ↑(c N) + ε * ↑(c (N - 1)) * l
[PROOFSTEP]
apply add_le_add
[GOAL]
case h₁
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ u (c N) - ↑(c N) * l ≤ ε * ↑(c N)
[PROOFSTEP]
apply (ha _ _).2
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ a ≤ N
[PROOFSTEP]
exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ a ≤ a + 1
[PROOFSTEP]
simp only [le_add_iff_nonneg_right, zero_le']
[GOAL]
case h₂
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ (↑(c N) - ↑(c (N - 1))) * l ≤ ε * ↑(c (N - 1)) * l
[PROOFSTEP]
apply mul_le_mul_of_nonneg_right _ lnonneg
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ↑(c N) - ↑(c (N - 1)) ≤ ε * ↑(c (N - 1))
[PROOFSTEP]
linarith only [IcN]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ε * ((1 + ε) * ↑(c (N - 1))) + ε * ↑(c (N - 1)) * l = ε * (1 + ε + l) * ↑(c (N - 1))
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ε * (1 + ε + l) * ↑(c (N - 1)) ≤ ε * (1 + ε + l) * ↑n
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ 0 ≤ ε * (1 + ε + l)
[PROOFSTEP]
apply mul_nonneg εpos.le
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, u (c n) - ↑(c n) * l ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ u (c b) - ↑(c b) * l ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
cNn : c (N - 1) ≤ n
IcN : ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
⊢ 0 ≤ 1 + ε + l
[PROOFSTEP]
linarith only [εpos, lnonneg]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
[PROOFSTEP]
have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (n : ℝ) * l - u n ≤ ε * (1 + l) * n :=
  by
  intro ε εpos
  rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
  have L : ∀ᶠ n : ℕ in atTop, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
    by
    rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
    filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
    have cnpos : 0 < c n := cnpos'
    calc
      (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
        simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
      _ ≤ ε * c n := by
        refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
        simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
        exact le_trans (neg_le_abs_self _) hn
  obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
    eventually_atTop.1 (cgrowth.and L)
  let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
  filter_upwards [Ici_mem_atTop M] with n hn
  have exN : ∃ N, n < c N := by
    rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
    exact ⟨N, by linarith only [hN]⟩
  let N := Nat.find exN
  have ncN : n < c N := Nat.find_spec exN
  have aN : a + 1 ≤ N := by
    by_contra' h
    have cNM : c N ≤ M := by
      apply le_max'
      apply mem_image_of_mem
      exact mem_range.2 h
    exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
  have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
  have aN' : a ≤ N - 1 := by
    apply @Nat.le_of_add_le_add_right a 1 (N - 1)
    rw [Nat.sub_add_cancel Npos]
    exact aN
  have cNn : c (N - 1) ≤ n := by
    have : N - 1 < N := Nat.pred_lt Npos.ne'
    simpa only [not_lt] using Nat.find_min exN this
  calc
    (n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) :=
      by
      refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _
      exact neg_le_neg (hmono cNn)
    _ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) :=
      by
      refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
      have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
      have := (ha _ aN').1
      rwa [B] at this
    _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
    _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
    _ = ε * (1 + l) * c (N - 1) := by ring
    _ ≤ ε * (1 + l) * n := by
      refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
      exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
⊢ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
intro ε εpos
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have L : ∀ᶠ n : ℕ in atTop, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
  by
  rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
  filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
  have cnpos : 0 < c n := cnpos'
  calc
    (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
      simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
    _ ≤ ε * c n := by
      refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
      simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
      exact le_trans (neg_le_abs_self _) hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
⊢ ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
[PROOFSTEP]
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ, Asymptotics.isLittleO_iff] at clim
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
⊢ ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
[PROOFSTEP]
filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
⊢ ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
[PROOFSTEP]
have cnpos : 0 < c n := cnpos'
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
[PROOFSTEP]
calc
  (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
    simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
  _ ≤ ε * c n := by
    refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
    simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
    exact le_trans (neg_le_abs_self _) hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ ↑(c n) * l - u (c n) = -(u (c n) / ↑(c n) - l) * ↑(c n)
[PROOFSTEP]
simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ -(u (c n) / ↑(c n) - l) * ↑(c n) ≤ ε * ↑(c n)
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
hn : ‖u (c n) / ↑(c n) - l‖ ≤ ε * ‖1‖
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
⊢ -(u (c n) / ↑(c n) - l) ≤ ε
[PROOFSTEP]
simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim✝ : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
clim : ∀ ⦃c_1 : ℝ⦄, 0 < c_1 → ∀ᶠ (x : ℕ) in atTop, ‖u (c x) / ↑(c x) - l‖ ≤ c_1 * ‖1‖
n : ℕ
cnpos' : n ∈ c ⁻¹' Set.Ioi 0
cnpos : 0 < c n
hn : |u (c n) / ↑(c n) - l| ≤ ε
⊢ -(u (c n) / ↑(c n) - l) ≤ ε
[PROOFSTEP]
exact le_trans (neg_le_abs_self _) hn
[GOAL]
case intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
⊢ ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
  eventually_atTop.1 (cgrowth.and L)
[GOAL]
case intro.intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
⊢ ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
⊢ Finset.Nonempty (image (fun i => c i) (range (a + 1)))
[PROOFSTEP]
simp
[GOAL]
case intro.intro.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
⊢ ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
filter_upwards [Ici_mem_atTop M] with n hn
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have exN : ∃ N, n < c N := by
  rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
  exact ⟨N, by linarith only [hN]⟩
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
⊢ ∃ N, n < c N
[PROOFSTEP]
rcases(tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
N : ℕ
hN : n + 1 ≤ c N
⊢ ∃ N, n < c N
[PROOFSTEP]
exact ⟨N, by linarith only [hN]⟩
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
N : ℕ
hN : n + 1 ≤ c N
⊢ n < c N
[PROOFSTEP]
linarith only [hN]
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
let N := Nat.find exN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have ncN : n < c N := Nat.find_spec exN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have aN : a + 1 ≤ N := by
  by_contra' h
  have cNM : c N ≤ M := by
    apply le_max'
    apply mem_image_of_mem
    exact mem_range.2 h
  exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
⊢ a + 1 ≤ N
[PROOFSTEP]
by_contra' h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ False
[PROOFSTEP]
have cNM : c N ≤ M := by
  apply le_max'
  apply mem_image_of_mem
  exact mem_range.2 h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ c N ≤ M
[PROOFSTEP]
apply le_max'
[GOAL]
case H2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ c N ∈ image (fun i => c i) (range (a + 1))
[PROOFSTEP]
apply mem_image_of_mem
[GOAL]
case H2.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
⊢ N ∈ range (a + 1)
[PROOFSTEP]
exact mem_range.2 h
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : Nat.find exN < a + 1
cNM : c N ≤ M
⊢ False
[PROOFSTEP]
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have aN' : a ≤ N - 1 := by
  apply @Nat.le_of_add_le_add_right a 1 (N - 1)
  rw [Nat.sub_add_cancel Npos]
  exact aN
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ a ≤ N - 1
[PROOFSTEP]
apply @Nat.le_of_add_le_add_right a 1 (N - 1)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ a + 1 ≤ N - 1 + 1
[PROOFSTEP]
rw [Nat.sub_add_cancel Npos]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
⊢ a + 1 ≤ N
[PROOFSTEP]
exact aN
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
have cNn : c (N - 1) ≤ n := by
  have : N - 1 < N := Nat.pred_lt Npos.ne'
  simpa only [not_lt] using Nat.find_min exN this
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
⊢ c (N - 1) ≤ n
[PROOFSTEP]
have : N - 1 < N := Nat.pred_lt Npos.ne'
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
this : N - 1 < N
⊢ c (N - 1) ≤ n
[PROOFSTEP]
simpa only [not_lt] using Nat.find_min exN this
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
calc
  (n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) :=
    by
    refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _
    exact neg_le_neg (hmono cNn)
  _ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) :=
    by
    refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
    have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
    have := (ha _ aN').1
    rwa [B] at this
  _ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
  _ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
  _ = ε * (1 + l) * c (N - 1) := by ring
  _ ≤ ε * (1 + l) * n := by
    refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
    exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ↑n * l - u n ≤ ↑(c N) * l - u (c (N - 1))
[PROOFSTEP]
refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ -u n ≤ -u (c (N - 1))
[PROOFSTEP]
exact neg_le_neg (hmono cNn)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ↑(c N) * l - u (c (N - 1)) ≤ (1 + ε) * ↑(c (N - 1)) * l - u (c (N - 1))
[PROOFSTEP]
refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
B : N - 1 + 1 = N
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
have := (ha _ aN').1
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
B : N - 1 + 1 = N
this : ↑(c (N - 1 + 1)) ≤ (1 + ε) * ↑(c (N - 1))
⊢ ↑(c N) ≤ (1 + ε) * ↑(c (N - 1))
[PROOFSTEP]
rwa [B] at this
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ (1 + ε) * ↑(c (N - 1)) * l - u (c (N - 1)) = ↑(c (N - 1)) * l - u (c (N - 1)) + ε * ↑(c (N - 1)) * l
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ε * ↑(c (N - 1)) + ε * ↑(c (N - 1)) * l = ε * (1 + l) * ↑(c (N - 1))
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ ε * (1 + l) * ↑(c (N - 1)) ≤ ε * (1 + l) * ↑n
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn :
  n ∈ Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
aN : a + 1 ≤ N
Npos : 0 < N
aN' : a ≤ N - 1
cNn : c (N - 1) ≤ n
⊢ 0 ≤ ε * (1 + l)
[PROOFSTEP]
exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
[PROOFSTEP]
refine' tendsto_order.2 ⟨fun d hd => _, fun d hd => _⟩
[GOAL]
case refine'_1
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
⊢ ∀ᶠ (b : ℕ) in atTop, d < u b / ↑b
[PROOFSTEP]
obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, d + ε * (1 + l) < l ∧ 0 < ε :=
  by
  have L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) :=
    by
    apply Tendsto.mono_left _ nhdsWithin_le_nhds
    exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
  simp only [zero_mul, add_zero] at L
  exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
⊢ ∃ ε, d + ε * (1 + l) < l ∧ 0 < ε
[PROOFSTEP]
have L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) :=
  by
  apply Tendsto.mono_left _ nhdsWithin_le_nhds
  exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
⊢ Tendsto (fun ε => d + ε * (1 + l)) (𝓝[Set.Ioi 0] 0) (𝓝 (d + 0 * (1 + l)))
[PROOFSTEP]
apply Tendsto.mono_left _ nhdsWithin_le_nhds
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
⊢ Tendsto (fun ε => d + ε * (1 + l)) (𝓝 0) (𝓝 (d + 0 * (1 + l)))
[PROOFSTEP]
exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[Set.Ioi 0] 0) (𝓝 (d + 0 * (1 + l)))
⊢ ∃ ε, d + ε * (1 + l) < l ∧ 0 < ε
[PROOFSTEP]
simp only [zero_mul, add_zero] at L
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[Set.Ioi 0] 0) (𝓝 d)
⊢ ∃ ε, d + ε * (1 + l) < l ∧ 0 < ε
[PROOFSTEP]
exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists
[GOAL]
case refine'_1.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
⊢ ∀ᶠ (b : ℕ) in atTop, d < u b / ↑b
[PROOFSTEP]
filter_upwards [B ε εpos, Ioi_mem_atTop 0] with n hn npos
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < u n / ↑n
[PROOFSTEP]
simp_rw [div_eq_inv_mul]
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < (↑n)⁻¹ * u n
[PROOFSTEP]
calc
  d < (n : ℝ)⁻¹ * n * (l - ε * (1 + l)) := by
    rw [inv_mul_cancel, one_mul]
    · linarith only [hε]
    · exact Nat.cast_ne_zero.2 (ne_of_gt npos)
  _ = (n : ℝ)⁻¹ * (n * l - ε * (1 + l) * n) := by ring
  _ ≤ (n : ℝ)⁻¹ * u n := by
    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
    linarith only [hn]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < (↑n)⁻¹ * ↑n * (l - ε * (1 + l))
[PROOFSTEP]
rw [inv_mul_cancel, one_mul]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < l - ε * (1 + l)
[PROOFSTEP]
linarith only [hε]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n ≠ 0
[PROOFSTEP]
exact Nat.cast_ne_zero.2 (ne_of_gt npos)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * ↑n * (l - ε * (1 + l)) = (↑n)⁻¹ * (↑n * l - ε * (1 + l) * ↑n)
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * (↑n * l - ε * (1 + l) * ↑n) ≤ (↑n)⁻¹ * u n
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n * l - ε * (1 + l) * ↑n ≤ u n
[PROOFSTEP]
linarith only [hn]
[GOAL]
case refine'_2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
[PROOFSTEP]
obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε :=
  by
  have L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) :=
    by
    apply Tendsto.mono_left _ nhdsWithin_le_nhds
    exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
  simp only [zero_mul, add_zero] at L
  exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
[PROOFSTEP]
have L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) :=
  by
  apply Tendsto.mono_left _ nhdsWithin_le_nhds
  exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[Set.Ioi 0] 0) (𝓝 (l + 0 * (1 + 0 + l)))
[PROOFSTEP]
apply Tendsto.mono_left _ nhdsWithin_le_nhds
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝 0) (𝓝 (l + 0 * (1 + 0 + l)))
[PROOFSTEP]
exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[Set.Ioi 0] 0) (𝓝 (l + 0 * (1 + 0 + l)))
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
[PROOFSTEP]
simp only [zero_mul, add_zero] at L
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[Set.Ioi 0] 0) (𝓝 l)
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
[PROOFSTEP]
exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
[GOAL]
case refine'_2.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
[PROOFSTEP]
filter_upwards [A ε εpos, Ioi_mem_atTop 0] with n hn npos
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ u n / ↑n < d
[PROOFSTEP]
simp_rw [div_eq_inv_mul]
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * u n < d
[PROOFSTEP]
calc
  (n : ℝ)⁻¹ * u n ≤ (n : ℝ)⁻¹ * (n * l + ε * (1 + ε + l) * n) :=
    by
    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
    linarith only [hn]
  _ = (n : ℝ)⁻¹ * n * (l + ε * (1 + ε + l)) := by ring
  _ < d := by
    rwa [inv_mul_cancel, one_mul]
    exact Nat.cast_ne_zero.2 (ne_of_gt npos)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * u n ≤ (↑n)⁻¹ * (↑n * l + ε * (1 + ε + l) * ↑n)
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ u n ≤ ↑n * l + ε * (1 + ε + l) * ↑n
[PROOFSTEP]
linarith only [hn]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * (↑n * l + ε * (1 + ε + l) * ↑n) = (↑n)⁻¹ * ↑n * (l + ε * (1 + ε + l))
[PROOFSTEP]
ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * ↑n * (l + ε * (1 + ε + l)) < d
[PROOFSTEP]
rwa [inv_mul_cancel, one_mul]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n ≠ 0
[PROOFSTEP]
exact Nat.cast_ne_zero.2 (ne_of_gt npos)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
⊢ Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
[PROOFSTEP]
apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
⊢ ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
[PROOFSTEP]
intro a ha
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
⊢ ∃ c,
    (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
      Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
[PROOFSTEP]
obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
⊢ ∃ c,
    (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
      Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
[PROOFSTEP]
refine'
  ⟨fun n => ⌊c k ^ n⌋₊, _, (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
[PROOFSTEP]
have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
  intro n
  refine' zero_lt_one.trans_le _
  simp only [Real.rpow_nat_cast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le (cone k).le n]
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
⊢ ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
[PROOFSTEP]
intro n
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
n : ℕ
⊢ 0 < ↑⌊c k ^ n⌋₊
[PROOFSTEP]
refine' zero_lt_one.trans_le _
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
n : ℕ
⊢ 1 ≤ ↑⌊c k ^ n⌋₊
[PROOFSTEP]
simp only [Real.rpow_nat_cast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le (cone k).le n]
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
[PROOFSTEP]
have A :
  Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / c k ^ (n + 1) * c k / (⌊c k ^ n⌋₊ / c k ^ n)) atTop
    (𝓝 (1 * c k / 1)) :=
  by
  refine' Tendsto.div (Tendsto.mul _ tendsto_const_nhds) _ one_ne_zero
  · refine' tendsto_nat_floor_div_atTop.comp _
    exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)
  · refine' tendsto_nat_floor_div_atTop.comp _
    exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
[PROOFSTEP]
refine' Tendsto.div (Tendsto.mul _ tendsto_const_nhds) _ one_ne_zero
[GOAL]
case refine'_1
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1)) atTop (𝓝 1)
[PROOFSTEP]
refine' tendsto_nat_floor_div_atTop.comp _
[GOAL]
case refine'_1
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ Tendsto (fun n => c k ^ (n + 1)) atTop atTop
[PROOFSTEP]
exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)
[GOAL]
case refine'_2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ Tendsto (fun n => ↑⌊c k ^ n⌋₊ / c k ^ n) atTop (𝓝 1)
[PROOFSTEP]
refine' tendsto_nat_floor_div_atTop.comp _
[GOAL]
case refine'_2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
⊢ Tendsto (fun n => c k ^ n) atTop atTop
[PROOFSTEP]
exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
[PROOFSTEP]
have B : Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) atTop (𝓝 (c k)) :=
  by
  simp only [one_mul, div_one] at A
  convert A using 1
  ext1 n
  field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']
  ring
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
⊢ Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
[PROOFSTEP]
simp only [one_mul, div_one] at A
[GOAL]
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (c k))
⊢ Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
[PROOFSTEP]
convert A using 1
[GOAL]
case h.e'_3
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (c k))
⊢ (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) = fun n =>
    ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)
[PROOFSTEP]
ext1 n
[GOAL]
case h.e'_3.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (c k))
n : ℕ
⊢ ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊ = ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)
[PROOFSTEP]
field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']
[GOAL]
case h.e'_3.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (c k))
n : ℕ
⊢ ↑⌊c k ^ (n + 1)⌋₊ * (c k ^ (n + 1) * ↑⌊c k ^ n⌋₊) = ↑⌊c k ^ (n + 1)⌋₊ * c k * c k ^ n * ↑⌊c k ^ n⌋₊
[PROOFSTEP]
ring
[GOAL]
case intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
B : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
⊢ ∀ᶠ (n : ℕ) in atTop, ↑((fun n => ⌊c k ^ n⌋₊) (n + 1)) ≤ a * ↑((fun n => ⌊c k ^ n⌋₊) n)
[PROOFSTEP]
filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
[GOAL]
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k / (↑⌊c k ^ n⌋₊ / c k ^ n)) atTop (𝓝 (1 * c k / 1))
B : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 (c k))
n : ℕ
hn : ↑⌊c k ^ (n + 1)⌋₊ / ↑⌊c k ^ n⌋₊ < a
⊢ ↑⌊c k ^ (n + 1)⌋₊ ≤ a * ↑⌊c k ^ n⌋₊
[PROOFSTEP]
exact (div_le_iff (H n)).1 hn.le
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
have cpos : 0 < c := zero_lt_one.trans hc
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
have A : (0 : ℝ) < c⁻¹ ^ 2 := sq_pos_of_pos (inv_pos.2 cpos)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
have B : c ^ 2 * ((1 : ℝ) - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹ :=
  by
  rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]
  swap
  · exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)
  have : c ^ 3 = c ^ 2 * c := by ring
  simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
  rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
  simpa using pow_le_pow hc.le one_le_two
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
[PROOFSTEP]
rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ c ^ 2 * (c - 1) ≤ c ^ 3 * (1 - c⁻¹ ^ 2)
N : ℕ j : ℝ hj : 0 < j c : ℝ hc : 1 < c cpos : 0 < c A : 0 < c⁻¹ ^ 2 ⊢ 0 < 1 - c⁻¹ ^ 2
[PROOFSTEP]
swap
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ 0 < 1 - c⁻¹ ^ 2
[PROOFSTEP]
exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ c ^ 2 * (c - 1) ≤ c ^ 3 * (1 - c⁻¹ ^ 2)
[PROOFSTEP]
have : c ^ 3 = c ^ 2 * c := by ring
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
⊢ c ^ 3 = c ^ 2 * c
[PROOFSTEP]
ring
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
this : c ^ 3 = c ^ 2 * c
⊢ c ^ 2 * (c - 1) ≤ c ^ 3 * (1 - c⁻¹ ^ 2)
[PROOFSTEP]
simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
this : c ^ 3 = c ^ 2 * c
⊢ c ^ 2 * c * (c ^ 2)⁻¹ ≤ c ^ 2
[PROOFSTEP]
rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
this : c ^ 3 = c ^ 2 * c
⊢ c ≤ c ^ 2
[PROOFSTEP]
simpa using pow_le_pow hc.le one_le_two
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
calc
  (∑ i in (range N).filter fun i => j < c ^ i, (1 : ℝ) / (c ^ i) ^ 2) ≤
      ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (1 : ℝ) / (c ^ i) ^ 2 :=
    by
    refine' sum_le_sum_of_subset_of_nonneg _ fun i _hi _hident => div_nonneg zero_le_one (sq_nonneg _)
    intro i hi
    simp only [mem_filter, mem_range] at hi
    simp only [hi.1, mem_Ico, and_true_iff]
    apply Nat.floor_le_of_le
    apply le_of_lt
    rw [div_lt_iff (Real.log_pos hc), ← Real.log_pow]
    exact Real.log_lt_log hj hi.2
  _ = ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i :=
    by
    congr 1 with i
    simp [← pow_mul, mul_comm]
  _ ≤ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / ((1 : ℝ) - c⁻¹ ^ 2) :=
    by
    apply geom_sum_Ico_le_of_lt_one (sq_nonneg _)
    rw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]
    exact inv_lt_one hc
  _ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / ((1 : ℝ) - c⁻¹ ^ 2) :=
    by
    apply div_le_div _ _ _ le_rfl
    · apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)
    · rw [← Real.rpow_nat_cast]
      apply Real.rpow_le_rpow_of_exponent_ge A
      · exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)
      · exact (Nat.sub_one_lt_floor _).le
    · simpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)
  _ = c ^ 2 * ((1 : ℝ) - c⁻¹ ^ 2)⁻¹ / j ^ 2 :=
    by
    have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = (1 : ℝ) / j ^ 2 :=
      by
      apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
      · rw [one_div]
        exact inv_pos.2 (sq_pos_of_pos hj)
      rw [Real.log_rpow A]
      simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]
      field_simp [(Real.log_pos hc).ne']
      ring
    rw [Real.rpow_sub A, I]
    have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'
    field_simp [hj.ne', (zero_lt_one.trans hc).ne']
    ring
  _ ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 := by
    apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
    exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ ∑ i in filter (fun i => j < c ^ i) (range N), 1 / (c ^ i) ^ 2 ≤
    ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, 1 / (c ^ i) ^ 2
[PROOFSTEP]
refine' sum_le_sum_of_subset_of_nonneg _ fun i _hi _hident => div_nonneg zero_le_one (sq_nonneg _)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ filter (fun i => j < c ^ i) (range N) ⊆ Ico ⌊Real.log j / Real.log c⌋₊ N
[PROOFSTEP]
intro i hi
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i ∈ filter (fun i => j < c ^ i) (range N)
⊢ i ∈ Ico ⌊Real.log j / Real.log c⌋₊ N
[PROOFSTEP]
simp only [mem_filter, mem_range] at hi
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i < N ∧ j < c ^ i
⊢ i ∈ Ico ⌊Real.log j / Real.log c⌋₊ N
[PROOFSTEP]
simp only [hi.1, mem_Ico, and_true_iff]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i < N ∧ j < c ^ i
⊢ ⌊Real.log j / Real.log c⌋₊ ≤ i
[PROOFSTEP]
apply Nat.floor_le_of_le
[GOAL]
case h
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i < N ∧ j < c ^ i
⊢ Real.log j / Real.log c ≤ ↑i
[PROOFSTEP]
apply le_of_lt
[GOAL]
case h.a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i < N ∧ j < c ^ i
⊢ Real.log j / Real.log c < ↑i
[PROOFSTEP]
rw [div_lt_iff (Real.log_pos hc), ← Real.log_pow]
[GOAL]
case h.a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
hi : i < N ∧ j < c ^ i
⊢ Real.log j < Real.log (c ^ i)
[PROOFSTEP]
exact Real.log_lt_log hj hi.2
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, 1 / (c ^ i) ^ 2 = ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i
[PROOFSTEP]
congr 1 with i
[GOAL]
case e_f.h
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
i : ℕ
⊢ 1 / (c ^ i) ^ 2 = (c⁻¹ ^ 2) ^ i
[PROOFSTEP]
simp [← pow_mul, mul_comm]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i ≤ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / (1 - c⁻¹ ^ 2)
[PROOFSTEP]
apply geom_sum_Ico_le_of_lt_one (sq_nonneg _)
[GOAL]
case h'x
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ c⁻¹ ^ 2 < 1
[PROOFSTEP]
rw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]
[GOAL]
case h'x
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ c⁻¹ < 1
[PROOFSTEP]
exact inv_lt_one hc
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / (1 - c⁻¹ ^ 2) ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c⁻¹ ^ 2)
[PROOFSTEP]
apply div_le_div _ _ _ le_rfl
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ 0 ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1)
[PROOFSTEP]
apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1)
[PROOFSTEP]
rw [← Real.rpow_nat_cast]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (c⁻¹ ^ 2) ^ ↑⌊Real.log j / Real.log c⌋₊ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1)
[PROOFSTEP]
apply Real.rpow_le_rpow_of_exponent_ge A
[GOAL]
case hx1
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ c⁻¹ ^ 2 ≤ 1
[PROOFSTEP]
exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)
[GOAL]
case hyz
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ Real.log j / Real.log c - 1 ≤ ↑⌊Real.log j / Real.log c⌋₊
[PROOFSTEP]
exact (Nat.sub_one_lt_floor _).le
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ 0 < 1 - c⁻¹ ^ 2
[PROOFSTEP]
simpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2
[PROOFSTEP]
have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = (1 : ℝ) / j ^ 2 :=
  by
  apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
  · rw [one_div]
    exact inv_pos.2 (sq_pos_of_pos hj)
  rw [Real.log_rpow A]
  simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]
  field_simp [(Real.log_pos hc).ne']
  ring
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
[PROOFSTEP]
apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ 1 / j ^ 2 ∈ Set.Ioi 0
[PROOFSTEP]
rw [one_div]
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ (j ^ 2)⁻¹ ∈ Set.Ioi 0
[PROOFSTEP]
exact inv_pos.2 (sq_pos_of_pos hj)
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ Real.log ((c⁻¹ ^ 2) ^ (Real.log j / Real.log c)) = Real.log (1 / j ^ 2)
[PROOFSTEP]
rw [Real.log_rpow A]
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ Real.log j / Real.log c * Real.log (c⁻¹ ^ 2) = Real.log (1 / j ^ 2)
[PROOFSTEP]
simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ Real.log j / Real.log c * (↑2 * Real.log c) = ↑2 * Real.log j
[PROOFSTEP]
field_simp [(Real.log_pos hc).ne']
[GOAL]
case a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ Real.log j * (2 * Real.log c) = 2 * Real.log j * Real.log c
[PROOFSTEP]
ring
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
⊢ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2
[PROOFSTEP]
rw [Real.rpow_sub A, I]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
⊢ 1 / j ^ 2 / (c⁻¹ ^ 2) ^ 1 / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2
[PROOFSTEP]
have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
this : c ^ 2 - 1 ≠ 0
⊢ 1 / j ^ 2 / (c⁻¹ ^ 2) ^ 1 / (1 - c⁻¹ ^ 2) = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2
[PROOFSTEP]
field_simp [hj.ne', (zero_lt_one.trans hc).ne']
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2
this : c ^ 2 - 1 ≠ 0
⊢ (c ^ 2 - 1) * j ^ 2 = j ^ 2 * (c ^ 2 - 1)
[PROOFSTEP]
ring
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < c⁻¹ ^ 2
B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹
⊢ 0 ≤ c ^ 3 * (c - 1)⁻¹
[PROOFSTEP]
exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
[GOAL]
c : ℝ
hc : 1 < c
i : ℕ
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
have cpos : 0 < c := zero_lt_one.trans hc
[GOAL]
c : ℝ
hc : 1 < c
i : ℕ
cpos : 0 < c
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
rcases Nat.eq_zero_or_pos i with (rfl | hi)
[GOAL]
case inl
c : ℝ
hc : 1 < c
cpos : 0 < c
⊢ (1 - c⁻¹) * c ^ 0 ≤ ↑⌊c ^ 0⌋₊
[PROOFSTEP]
simp only [pow_zero, Nat.floor_one, Nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg, cpos.le]
[GOAL]
case inr
c : ℝ
hc : 1 < c
i : ℕ
cpos : 0 < c
hi : i > 0
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
have hident : 1 ≤ i := hi
[GOAL]
case inr
c : ℝ
hc : 1 < c
i : ℕ
cpos : 0 < c
hi : i > 0
hident : 1 ≤ i
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
calc
  (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
  _ ≤ c ^ i - 1 := by
    simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using pow_le_pow hc.le hident
  _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
[GOAL]
c : ℝ
hc : 1 < c
i : ℕ
cpos : 0 < c
hi : i > 0
hident : 1 ≤ i
⊢ (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹
[PROOFSTEP]
ring
[GOAL]
c : ℝ
hc : 1 < c
i : ℕ
cpos : 0 < c
hi : i > 0
hident : 1 ≤ i
⊢ c ^ i - c ^ i * c⁻¹ ≤ c ^ i - 1
[PROOFSTEP]
simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using pow_le_pow hc.le hident
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
⊢ ∑ i in filter (fun x => j < ↑⌊c ^ x⌋₊) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
[PROOFSTEP]
have cpos : 0 < c := zero_lt_one.trans hc
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
⊢ ∑ i in filter (fun x => j < ↑⌊c ^ x⌋₊) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
[PROOFSTEP]
have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one hc)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∑ i in filter (fun x => j < ↑⌊c ^ x⌋₊) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
[PROOFSTEP]
calc
  (∑ i in (range N).filter (j < ⌊c ^ ·⌋₊), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
      ∑ i in (range N).filter (j < c ^ ·), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2 :=
    by
    apply sum_le_sum_of_subset_of_nonneg
    · intro i hi
      simp only [mem_filter, mem_range] at hi
      simpa only [hi.1, mem_filter, mem_range, true_and_iff] using hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
    · intro i _hi _hident
      exact div_nonneg zero_le_one (sq_nonneg _)
  _ ≤ ∑ i in (range N).filter (j < c ^ ·), (1 - c⁻¹)⁻¹ ^ 2 * ((1 : ℝ) / (c ^ i) ^ 2) :=
    by
    refine' sum_le_sum fun i _hi => _
    rw [mul_div_assoc', mul_one, div_le_div_iff]; rotate_left
    · apply sq_pos_of_pos
      refine' zero_lt_one.trans_le _
      simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
    · exact sq_pos_of_pos (pow_pos cpos _)
    rw [one_mul, ← mul_pow]
    apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)
    rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
    exact mul_pow_le_nat_floor_pow hc i
  _ ≤ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 :=
    by
    rw [← mul_sum, ← mul_div_assoc']
    refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)
    exact sum_div_pow_sq_le_div_sq N hj hc
  _ = c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 := by
    congr 1
    field_simp [cpos.ne', (sub_pos.2 hc).ne']
    ring!
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∑ i in filter (fun x => j < ↑⌊c ^ x⌋₊) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤
    ∑ i in filter (fun x => j < c ^ x) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2
[PROOFSTEP]
apply sum_le_sum_of_subset_of_nonneg
[GOAL]
case h
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ filter (fun x => j < ↑⌊c ^ x⌋₊) (range N) ⊆ filter (fun x => j < c ^ x) (range N)
[PROOFSTEP]
intro i hi
[GOAL]
case h
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
hi : i ∈ filter (fun x => j < ↑⌊c ^ x⌋₊) (range N)
⊢ i ∈ filter (fun x => j < c ^ x) (range N)
[PROOFSTEP]
simp only [mem_filter, mem_range] at hi
[GOAL]
case h
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
hi : i < N ∧ j < ↑⌊c ^ i⌋₊
⊢ i ∈ filter (fun x => j < c ^ x) (range N)
[PROOFSTEP]
simpa only [hi.1, mem_filter, mem_range, true_and_iff] using hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
[GOAL]
case hf
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∀ (i : ℕ),
    i ∈ filter (fun x => j < c ^ x) (range N) → ¬i ∈ filter (fun x => j < ↑⌊c ^ x⌋₊) (range N) → 0 ≤ 1 / ↑⌊c ^ i⌋₊ ^ 2
[PROOFSTEP]
intro i _hi _hident
[GOAL]
case hf
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
_hident : ¬i ∈ filter (fun x => j < ↑⌊c ^ x⌋₊) (range N)
⊢ 0 ≤ 1 / ↑⌊c ^ i⌋₊ ^ 2
[PROOFSTEP]
exact div_nonneg zero_le_one (sq_nonneg _)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤
    ∑ i in filter (fun x => j < c ^ x) (range N), (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)
[PROOFSTEP]
refine' sum_le_sum fun i _hi => _
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)
[PROOFSTEP]
rw [mul_div_assoc', mul_one, div_le_div_iff]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 1 * (c ^ i) ^ 2 ≤ (1 - c⁻¹)⁻¹ ^ 2 * ↑⌊c ^ i⌋₊ ^ 2
case b0
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 0 < ↑⌊c ^ i⌋₊ ^ 2
case d0
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 0 < (c ^ i) ^ 2
[PROOFSTEP]
rotate_left
[GOAL]
case b0
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 0 < ↑⌊c ^ i⌋₊ ^ 2
[PROOFSTEP]
apply sq_pos_of_pos
[GOAL]
case b0.ha
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 0 < ↑⌊c ^ i⌋₊
[PROOFSTEP]
refine' zero_lt_one.trans_le _
[GOAL]
case b0.ha
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 1 ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
[GOAL]
case d0
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 0 < (c ^ i) ^ 2
[PROOFSTEP]
exact sq_pos_of_pos (pow_pos cpos _)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ 1 * (c ^ i) ^ 2 ≤ (1 - c⁻¹)⁻¹ ^ 2 * ↑⌊c ^ i⌋₊ ^ 2
[PROOFSTEP]
rw [one_mul, ← mul_pow]
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ (c ^ i) ^ 2 ≤ ((1 - c⁻¹)⁻¹ * ↑⌊c ^ i⌋₊) ^ 2
[PROOFSTEP]
apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)
[GOAL]
case hab
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ c ^ i ≤ (1 - c⁻¹)⁻¹ * ↑⌊c ^ i⌋₊
[PROOFSTEP]
rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
[GOAL]
case hab
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
_hi : i ∈ filter (fun x => j < c ^ x) (range N)
⊢ (1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊
[PROOFSTEP]
exact mul_pow_le_nat_floor_pow hc i
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∑ i in filter (fun x => j < c ^ x) (range N), (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2) ≤
    (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2
[PROOFSTEP]
rw [← mul_sum, ← mul_div_assoc']
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ (1 - c⁻¹)⁻¹ ^ 2 * ∑ x in filter (fun x => j < c ^ x) (range N), 1 / (c ^ x) ^ 2 ≤
    (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹ / j ^ 2)
[PROOFSTEP]
refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ ∑ x in filter (fun x => j < c ^ x) (range N), 1 / (c ^ x) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2
[PROOFSTEP]
exact sum_div_pow_sq_le_div_sq N hj hc
[GOAL]
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 = c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2
[PROOFSTEP]
congr 1
[GOAL]
case e_a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) = c ^ 5 * (c - 1)⁻¹ ^ 3
[PROOFSTEP]
field_simp [cpos.ne', (sub_pos.2 hc).ne']
[GOAL]
case e_a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ c ^ 2 * c ^ 3 * (c - 1) ^ 3 = c ^ 5 * ((c - 1) ^ 2 * (c - 1))
[PROOFSTEP]
ring!