-- 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:
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 := (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
⊢ c N ∈ image (fun i => c i) (range (a + 1))

Tactic:
apply mem_image_of_mem

Result:
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 := (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
⊢ N ∈ range (a + 1)

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 : 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 := (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
cNn : c (N - 1) ≤ n
⊢ a ≤ N - 1

Tactic:
apply @Nat.le_of_add_le_add_right a 1 (N - 1)

Result:
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 := (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
cNn : c (N - 1) ≤ n
⊢ a + 1 ≤ N - 1 + 1

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
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

Tactic:
calc
  (n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) := by
    gcongr
    exact hmono cNn
  _ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) := by
    gcongr
    have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
    simpa [B] using (ha _ aN').1
  _ = 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 gcongr

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)
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)

Tactic:
gcongr

Result:
case hab
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 ≤ u (c 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
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
⊢ ε * (1 + l) * ↑(c (N - 1)) ≤ ε * (1 + l) * ↑n

Tactic:
gcongr

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
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 : 0 < n
⊢ ↑n * l + ε * (1 + ε + l) * ↑n = (l + ε * (1 + ε + l)) * ↑n

Tactic:
ring

Result:
no goals