-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
    (hmono : Monotone u)
    (hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ 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)

Proof state:
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 : ∀ ⦃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)

Tactic:
simp only [cnpos.ne', Ne, Nat.cast_eq_zero, not_false_iff, field_simps]

Result:
no goals

Proof state:
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 := (image (fun i => c i) (range (a + 1))).max' ⋯
n : ℕ
hn : n ∈ Set.Ici M
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : N < a + 1
cNM : c N ≤ M
⊢ False

Tactic:
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)

Result:
no goals

Proof state:
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 := (image (fun i => c i) (range (a + 1))).max' ⋯
n : ℕ
hn : n ∈ Set.Ici M
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n

Tactic:
have ncN : n < c N := Nat.find_spec exN

Result:
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 := (image (fun i => c i) (range (a + 1))).max' ⋯
n : ℕ
hn : n ∈ Set.Ici M
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
⊢ ↑n * l - u n ≤ ε * (1 + l) * ↑n

Proof state:
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)

Tactic:
refine tendsto_order.2 ⟨fun d hd => ?_, fun d hd => ?_⟩

Result:
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

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

Proof state:
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 := (image (fun i => c i) (range (a + 1))).max' ⋯
n : ℕ
hn : n ∈ Set.Ici M
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))

Tactic:
gcongr

Result:
case h.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 := (image (fun i => c i) (range (a + 1))).max' ⋯
n : ℕ
hn : n ∈ Set.Ici M
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))