l3lab/ntp-mathlib huggingface dataset:
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
H : 0 < (2 * (↑n + 1) + 1) ^ 2 - 1
⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 < (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2

Next tactic:
convert H using 1
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> 

Proof state:
case h.e'_3
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
H : 0 < (2 * (↑n + 1) + 1) ^ 2 - 1
⊢ 0 * (2 * (↑n + 1) + 1) ^ 2 = 0

Next tactic:
field_simp [h₃.ne']
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> 

Proof state:
case h.e'_4
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
H : 0 < (2 * (↑n + 1) + 1) ^ 2 - 1
⊢ (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) * (2 * (↑n + 1) + 1) ^ 2 = (2 * (↑n + 1) + 1) ^ 2 - 1

Next tactic:
field_simp [h₃.ne']
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ Real.log (stirlingSeq (n + 1)) - Real.log (stirlingSeq (n + 2)) ≤ 1 / (4 * ↑(n + 1) ^ 2)

Next tactic:
refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ (1 / (2 * ↑(n + 1) + 1)) ^ 2 / (1 - (1 / (2 * ↑(n + 1) + 1)) ^ 2) ≤ 1 / (4 * ↑(n + 1) ^ 2)

Next tactic:
push_cast
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 / (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2) ≤ 1 / (4 * (↑n + 1) ^ 2)

Next tactic:
rw [div_le_div_iff h₂ h₁]
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  rw [div_le_div_iff h₂ h₁]
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ (1 / (2 * (↑n + 1) + 1)) ^ 2 * (4 * (↑n + 1) ^ 2) ≤ 1 * (1 - (1 / (2 * (↑n + 1) + 1)) ^ 2)

Next tactic:
field_simp [h₃.ne']
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  rw [div_le_div_iff h₂ h₁]
  field_simp [h₃.ne']
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ 4 * (↑n + 1) ^ 2 / (2 * (↑n + 1) + 1) ^ 2 ≤ ((2 * (↑n + 1) + 1) ^ 2 - 1) / (2 * (↑n + 1) + 1) ^ 2

Next tactic:
rw [div_le_div_right h₃]
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  rw [div_le_div_iff h₂ h₁]
  field_simp [h₃.ne']
  rw [div_le_div_right h₃]
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ 4 * (↑n + 1) ^ 2 ≤ (2 * (↑n + 1) + 1) ^ 2 - 1

Next tactic:
ring_nf
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  rw [div_le_div_iff h₂ h₁]
  field_simp [h₃.ne']
  rw [div_le_div_right h₃]
  ring_nf
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ 4 + ↑n * 8 + ↑n ^ 2 * 4 ≤ 8 + ↑n * 12 + ↑n ^ 2 * 4

Next tactic:
norm_cast
/-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
-/
theorem log_stirlingSeq_sub_log_stirlingSeq_succ (n : ℕ) :
    log (stirlingSeq (n + 1)) - log (stirlingSeq (n + 2)) ≤ 1 / (4 * (↑(n + 1):ℝ) ^ 2) := by
  have h₁ : (0 : ℝ) < ↑4 * ((n:ℝ) + 1) ^ 2 := by positivity
  have h₃ : (0 : ℝ) < (2 * ((n:ℝ) + 1) + 1) ^ 2 := by positivity
  have h₂ : (0 : ℝ) < ↑1 - (1 / (2 * ((n:ℝ) + 1) + 1)) ^ 2 := by
    rw [← mul_lt_mul_right h₃]
    have H : ↑0 < (2 * ((n:ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n]
    convert H using 1 <;> field_simp [h₃.ne']
  refine' (log_stirlingSeq_diff_le_geo_sum n).trans _
  push_cast
  rw [div_le_div_iff h₂ h₁]
  field_simp [h₃.ne']
  rw [div_le_div_right h₃]
  ring_nf
  norm_cast
  

Proof state:
n : ℕ
h₁ : 0 < 4 * (↑n + 1) ^ 2
h₃ : 0 < (2 * (↑n + 1) + 1) ^ 2
h₂ : 0 < 1 - (1 / (2 * (↑n + 1) + 1)) ^ 2
⊢ 4 + n * 8 + n ^ 2 * 4 ≤ 8 + n * 12 + n ^ 2 * 4

Next tactic:
linarith
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  

Proof state:
⊢ ∃ c, ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ c

Next tactic:
let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
⊢ ∃ c, ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ c

Next tactic:
use 1 / 4 * d
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  

Proof state:
case h
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
⊢ ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ 1 / 4 * d

Next tactic:
let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  

Proof state:
case h
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
⊢ ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ 1 / 4 * d

Next tactic:
intro n
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  

Proof state:
case h
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
⊢ Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ 1 / 4 * d

Next tactic:
have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
⊢ ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)

Next tactic:
intro k
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; 

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n k : ℕ
⊢ log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)

Next tactic:
convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; 

Proof state:
case h.e'_4
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n k : ℕ
⊢ 1 / 4 * (1 / ↑(k + 1) ^ 2) = 1 / (4 * ↑(k + 1) ^ 2)

Next tactic:
field_simp
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  

Proof state:
case h
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
⊢ Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ 1 / 4 * d

Next tactic:
have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    simp only [rpow_nat_cast] at this
    exact sum_le_tsum (range n) (fun k _ => by positivity) this
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
⊢ ∑ k in range n, 1 / ↑(k + 1) ^ 2 ≤ d

Next tactic:
have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
this : Summable fun n => 1 / ↑(n + 1) ^ 2
⊢ ∑ k in range n, 1 / ↑(k + 1) ^ 2 ≤ d

Next tactic:
simp only [rpow_nat_cast] at this
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    simp only [rpow_nat_cast] at this
    

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
this : Summable fun n => 1 / ↑(n + 1) ^ 2
⊢ ∑ k in range n, 1 / ↑(k + 1) ^ 2 ≤ d

Next tactic:
exact sum_le_tsum (range n) (fun k _ => by positivity) this
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    simp only [rpow_nat_cast] at this
    exact sum_le_tsum (range n) (fun k _ => by 

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
this : Summable fun n => 1 / ↑(n + 1) ^ 2
k : ℕ
x✝ : k ∉ range n
⊢ 0 ≤ 1 / ↑(k + 1) ^ 2

Next tactic:
positivity
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    simp only [rpow_nat_cast] at this
    exact sum_le_tsum (range n) (fun k _ => by positivity) this
  

Proof state:
case h
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
h₂ : ∑ k in range n, 1 / ↑(k + 1) ^ 2 ≤ d
⊢ Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ 1 / 4 * d

Next tactic:
calc
    log (stirlingSeq 1) - log (stirlingSeq (n + 1)) = log_stirlingSeq' 0 - log_stirlingSeq' n :=
      rfl
    _ = ∑ k in range n, (log_stirlingSeq' k - log_stirlingSeq' (k + 1)) := by
      rw [← sum_range_sub' log_stirlingSeq' n]
    _ ≤ ∑ k in range n, ↑1 / ↑4 * (↑1 / ↑((k + 1)) ^ 2) := (sum_le_sum fun k _ => h₁ k)
    _ = ↑1 / ↑4 * ∑ k in range n, ↑1 / ↑((k + 1)) ^ 2 := by rw [mul_sum]
    _ ≤ 1 / 4 * d := by gcongr
/-- For any `n`, we have `log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2`  -/
theorem log_stirlingSeq_bounded_aux :
    ∃ c : ℝ, ∀ n : ℕ, log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ c := by
  let d : ℝ := ∑' k : ℕ, (1 : ℝ) / (↑(k + 1) : ℝ) ^ 2
  use 1 / 4 * d
  let log_stirlingSeq' : ℕ → ℝ := fun k => log (stirlingSeq (k + 1))
  intro n
  have h₁ : ∀ k, log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤
      ↑1 / ↑4 * (↑1 / (↑(k + 1):ℝ) ^ 2) := by
    intro k; convert log_stirlingSeq_sub_log_stirlingSeq_succ k using 1; field_simp
  have h₂ : (∑ k : ℕ in range n, ↑1 / (↑(k + 1):ℝ) ^ 2) ≤ d := by
    have := (summable_nat_add_iff 1).mpr <| Real.summable_one_div_nat_pow.mpr one_lt_two
    simp only [rpow_nat_cast] at this
    exact sum_le_tsum (range n) (fun k _ => by positivity) this
  calc
    log (stirlingSeq 1) - log (stirlingSeq (n + 1)) = log_stirlingSeq' 0 - log_stirlingSeq' n :=
      rfl
    _ = ∑ k in range n, (log_stirlingSeq' k - log_stirlingSeq' (k + 1)) := by
      

Proof state:
d : ℝ := ∑' (k : ℕ), 1 / ↑(k + 1) ^ 2
log_stirlingSeq' : ℕ → ℝ := fun k => Real.log (stirlingSeq (k + 1))
n : ℕ
h₁ : ∀ (k : ℕ), log_stirlingSeq' k - log_stirlingSeq' (k + 1) ≤ 1 / 4 * (1 / ↑(k + 1) ^ 2)
h₂ : ∑ k in range n, 1 / ↑(k + 1) ^ 2 ≤ d
⊢ log_stirlingSeq' 0 - log_stirlingSeq' n = ∑ k in range n, (log_stirlingSeq' k - log_stirlingSeq' (k + 1))

Next tactic:
rw [← sum_range_sub' log_stirlingSeq' n]