Theorem: tendsto_div_of_monotone_of_tendsto_div_floor_pow
File: Mathlib/Analysis/SpecificLimits/FloorPow.lean

Proof steps:
state_before: 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)
state_after: 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)
tactic: apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
state_before: 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)
state_after: 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)
tactic: intro a ha
state_before: 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)
state_after: 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)
tactic: obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
state_before: 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)
state_after: 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)
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⟩
state_before: 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)
state_after: 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)
tactic: have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
    intro n
    refine zero_lt_one.trans_le ?_
    simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff,
      one_le_pow_of_one_le (cone k).le n]
state_before: 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)
state_after: 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)
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
state_before: 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)
state_after: 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⌋₊
tactic: filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
state_before: 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⌋₊
state_after: no goals
tactic: exact (div_le_iff (H n)).1 hn.le