l3lab/ntp-mathlib huggingface dataset:
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  

Proof state:
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)

Next tactic:
intro a ha
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  

Proof state:
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)

Next tactic:
obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  

Proof state:
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)

Next tactic:
refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  

Proof state:
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)

Next tactic:
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]
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
    

Proof state:
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⌋₊

Next tactic:
intro n
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
    intro n
    

Proof state:
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⌋₊

Next tactic:
refine' zero_lt_one.trans_le _
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
    intro n
    refine' zero_lt_one.trans_le _
    

Proof state:
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⌋₊

Next tactic:
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]
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  

Proof state:
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)

Next tactic:
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)
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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
    

Proof state:
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))

Next tactic:
refine' Tendsto.div (Tendsto.mul _ tendsto_const_nhds) _ one_ne_zero
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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
    · 

Proof state:
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)

Next tactic:
refine' tendsto_nat_floor_div_atTop.comp _
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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 _
      

Proof state:
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

Next tactic:
exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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)
    · 

Proof state:
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)

Next tactic:
refine' tendsto_nat_floor_div_atTop.comp _
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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 _
      

Proof state:
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

Next tactic:
exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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)
  

Proof state:
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)

Next tactic:
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
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
    (c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
    (hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
  intro a ha
  obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
  refine'
    ⟨fun n => ⌊c k ^ n⌋₊, _,
      (tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
  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]
  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)
  have B : Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) atTop (𝓝 (c k)) := by
    

Proof state:
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))

Next tactic:
simp only [one_mul, div_one] at A