-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean
theorem tendsto_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
    (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) :
    Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1)

Proof state:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
have :
  ∀ᶠ n : ℕ in atTop,
    logGammaSeq (x - 1) n =
      logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1) := by
  refine Eventually.mp (eventually_ge_atTop 1) (Eventually.of_forall fun n hn => ?_)
  have := logGammaSeq_add_one (x - 1) (n - 1)
  rw [sub_add_cancel, Nat.sub_add_cancel hn] at this
  rw [this]
  ring

Result:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
this :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Proof state:
case h
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm :
  Tendsto
    ((fun x_1 => logGammaSeq x (x_1 - 1) + x * (log (↑(x_1 - 1) + 1) - log ↑(x_1 - 1)) - log (x - 1) + log (x - 1)) ∘
      fun a => a + 1)
    atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
n : ℕ
⊢ logGammaSeq x (n + 1 - 1) + x * (log (↑(n + 1 - 1) + 1) - log ↑(n + 1 - 1)) - log (x - 1) + log (x - 1) =
    logGammaSeq x n + x * (log (↑n + 1) - log ↑n)

Tactic:
rw [sub_add_cancel, Nat.add_sub_cancel]

Result:
no goals

Proof state:
case succ
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 =>
        (fun n => logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b => log (x - 1)) x_1) ∘
      fun a => a + 1) =
    fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))

Tactic:
convert hm.sub (tendsto_log_nat_add_one_sub_log.const_mul x) using 2

Result:
case h.e'_3.h
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 =>
        (fun n => logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b => log (x - 1)) x_1) ∘
      fun a => a + 1) =
    fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
x✝ : ℕ
⊢ logGammaSeq x x✝ = logGammaSeq x x✝ + x * (log (↑x✝ + 1) - log ↑x✝) - x * (log (↑x✝ + 1) - log ↑x✝)

case h.e'_5.h.e'_3
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 =>
        (fun n => logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b => log (x - 1)) x_1) ∘
      fun a => a + 1) =
    fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0

Proof state:
case h.e'_5.h.e'_3
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this :
  ((fun x_1 =>
        (fun n => logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b => log (x - 1)) x_1) ∘
      fun a => a + 1) =
    fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0

Tactic:
have := hf_feq ((Nat.cast_nonneg m).trans_lt hy)

Result:
case h.e'_5.h.e'_3
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
this✝¹ :
  ∀ᶠ (n : ℕ) in atTop,
    logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)
hm : Tendsto (fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)) atTop (𝓝 (f (x - 1) - f 1 + log (x - 1)))
this✝ :
  ((fun x_1 =>
        (fun n => logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)) x_1 +
          (fun b => log (x - 1)) x_1) ∘
      fun a => a + 1) =
    fun n => logGammaSeq x n + x * (log (↑n + 1) - log ↑n)
this : f (x - 1 + 1) = f (x - 1) + log (x - 1)
⊢ f x - f 1 = f (x - 1) - f 1 + log (x - 1) - x * 0

Proof state:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n = logGammaSeq x (n - 1) + x * (log (↑(n - 1) + 1) - log ↑(n - 1)) - log (x - 1)

Tactic:
rw [this]

Result:
f : ℝ → ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
m : ℕ
x : ℝ
hx : 0 < x
hy : ↑m < x - 1
hy' : x - 1 ≤ ↑m + 1
hm : Tendsto (logGammaSeq (x - 1)) atTop (𝓝 (f (x - 1) - f 1))
n : ℕ
hn : 1 ≤ n
this : logGammaSeq x (n - 1) = logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1))
⊢ logGammaSeq (x - 1) n =
    logGammaSeq (x - 1) n + log (x - 1) - x * (log (↑(n - 1) + 1) - log ↑(n - 1)) +
        x * (log (↑(n - 1) + 1) - log ↑(n - 1)) -
      log (x - 1)