File: Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean

full_name: Real.eulerMascheroniSeq
code: /-- The sequence with `n`-th term `harmonic n - log (n + 1)`. -/
noncomputable def eulerMascheroniSeq (n : ℕ) : ℝ 
goal: ℝ

full_name: Real.eulerMascheroniSeq_zero
code: lemma eulerMascheroniSeq_zero : eulerMascheroniSeq 0 = 0 
goal: Real.eulerMascheroniSeq 0 = 0

full_name: Real.strictMono_eulerMascheroniSeq
code: lemma strictMono_eulerMascheroniSeq : StrictMono eulerMascheroniSeq 
goal: StrictMono Real.eulerMascheroniSeq

full_name: Real.one_half_lt_eulerMascheroniSeq_six
code: lemma one_half_lt_eulerMascheroniSeq_six : 1 / 2 < eulerMascheroniSeq 6 
goal: 1 / 2 < Real.eulerMascheroniSeq 6

full_name: Real.eulerMascheroniSeq'
code: /-- The sequence with `n`-th term `harmonic n - log n`. We use a junk value for `n = 0`, in order
to have the sequence be strictly decreasing. -/
noncomputable def eulerMascheroniSeq' (n : ℕ) : ℝ 
goal: ℝ

full_name: Real.eulerMascheroniSeq'_one
code: lemma eulerMascheroniSeq'_one : eulerMascheroniSeq' 1 = 1 
goal: Real.eulerMascheroniSeq' 1 = 1

full_name: Real.strictAnti_eulerMascheroniSeq'
code: lemma strictAnti_eulerMascheroniSeq' : StrictAnti eulerMascheroniSeq' 
goal: StrictAnti Real.eulerMascheroniSeq'

full_name: Real.eulerMascheroniSeq'_six_lt_two_thirds
code: lemma eulerMascheroniSeq'_six_lt_two_thirds : eulerMascheroniSeq' 6 < 2 / 3 
goal: Real.eulerMascheroniSeq' 6 < 2 / 3

full_name: Real.eulerMascheroniSeq_lt_eulerMascheroniSeq'
code: lemma eulerMascheroniSeq_lt_eulerMascheroniSeq' (m n : ℕ) :
    eulerMascheroniSeq m < eulerMascheroniSeq' n 
goal: Real.eulerMascheroniSeq m < Real.eulerMascheroniSeq' n

full_name: Real.eulerMascheroniConstant
code: /-- The Euler-Mascheroni constant `γ`. -/
noncomputable def eulerMascheroniConstant : ℝ 
goal: ℝ

full_name: Real.tendsto_eulerMascheroniSeq
code: lemma tendsto_eulerMascheroniSeq :
    Tendsto eulerMascheroniSeq atTop (𝓝 eulerMascheroniConstant) 
goal: Filter.Tendsto Real.eulerMascheroniSeq Filter.atTop (𝓝 Real.eulerMascheroniConstant)

full_name: Real.tendsto_harmonic_sub_log_add_one
code: lemma tendsto_harmonic_sub_log_add_one :
    Tendsto (fun n : ℕ ↦ harmonic n - log (n + 1)) atTop (𝓝 eulerMascheroniConstant) 
goal: Filter.Tendsto (fun n => ↑(harmonic n) - Real.log (↑n + 1)) Filter.atTop (𝓝 Real.eulerMascheroniConstant)

full_name: Real.tendsto_eulerMascheroniSeq'
code: lemma tendsto_eulerMascheroniSeq' :
    Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant) 
goal: Filter.Tendsto Real.eulerMascheroniSeq' Filter.atTop (𝓝 Real.eulerMascheroniConstant)

full_name: Real.tendsto_harmonic_sub_log
code: lemma tendsto_harmonic_sub_log :
    Tendsto (fun n : ℕ ↦ harmonic n - log n) atTop (𝓝 eulerMascheroniConstant) 
goal: Filter.Tendsto (fun n => ↑(harmonic n) - Real.log ↑n) Filter.atTop (𝓝 Real.eulerMascheroniConstant)

full_name: Real.eulerMascheroniSeq_lt_eulerMascheroniConstant
code: lemma eulerMascheroniSeq_lt_eulerMascheroniConstant (n : ℕ) :
    eulerMascheroniSeq n < eulerMascheroniConstant 
goal: Real.eulerMascheroniSeq n < Real.eulerMascheroniConstant

full_name: Real.eulerMascheroniConstant_lt_eulerMascheroniSeq'
code: lemma eulerMascheroniConstant_lt_eulerMascheroniSeq' (n : ℕ) :
    eulerMascheroniConstant < eulerMascheroniSeq' n 
goal: Real.eulerMascheroniConstant < Real.eulerMascheroniSeq' n

full_name: Real.one_half_lt_eulerMascheroniConstant
code: /-- Lower bound for `γ`. (The true value is about 0.57.) -/
lemma one_half_lt_eulerMascheroniConstant : 1 / 2 < eulerMascheroniConstant 
goal: 1 / 2 < Real.eulerMascheroniConstant

full_name: Real.eulerMascheroniConstant_lt_two_thirds
code: /-- Upper bound for `γ`. (The true value is about 0.57.) -/
lemma eulerMascheroniConstant_lt_two_thirds : eulerMascheroniConstant < 2 / 3 
goal: Real.eulerMascheroniConstant < 2 / 3