-- Mathlib/Topology/MetricSpace/Cauchy.lean
lemma Metric.exists_subseq_bounded_of_cauchySeq (u : ℕ → α) (hu : CauchySeq u) (b : ℕ → ℝ)
    (hb : ∀ n, 0 < b n) :
    ∃ f : ℕ → ℕ, StrictMono f ∧ ∀ n, ∀ m ≥ f n, dist (u m) (u (f n)) < b n

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n

Tactic:
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
  intro k
  rw [eventually_atTop]
  obtain ⟨N, hN⟩ := hu (b k) (hb k)
  exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩

Result:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
hu' : ∀ (k : ℕ), ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k
⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
hu' : ∀ (k : ℕ), ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k
⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n

Tactic:
exact Filter.extraction_forall_of_eventually hu'

Result:
no goals

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n

Tactic:
rw [cauchySeq_iff] at hu

Result:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k

Tactic:
rw [eventually_atTop]

Result:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∃ a, ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k

Proof state:
case intro
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < b k
⊢ ∃ a, ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k

Tactic:
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩

Result:
no goals

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
⊢ ∀ (k : ℕ), ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k

Tactic:
intro k

Result:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k

Proof state:
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k : ℕ
⊢ ∃ a, ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k

Tactic:
obtain ⟨N, hN⟩ := hu (b k) (hb k)

Result:
case intro
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε
b : ℕ → ℝ
hb : ∀ (n : ℕ), 0 < b n
k N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < b k
⊢ ∃ a, ∀ b_1 ≥ a, ∀ m ≥ b_1, dist (u m) (u b_1) < b k