-- Mathlib/Topology/UniformSpace/Cauchy.lean
theorem CauchySeq.subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
    (hu : CauchySeq u) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V n

Proof state:
case intro.intro
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
this : ∀ (n : ℕ), ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n l : ℕ), l ≥ φ n → (u l, u (φ n)) ∈ V n
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n

Tactic:
exact ⟨φ, φ_extr, fun n => hφ _ _ (φ_extr <| Nat.lt_add_one n).le⟩

Result:
no goals

Proof state:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V
n : ℕ
⊢ ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n

Tactic:
rcases hu _ (hV n) with ⟨N, H⟩

Result:
case intro
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V
n N : ℕ
H : ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V n
⊢ ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n

Proof state:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
this : ∀ (n : ℕ), ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n

Tactic:
obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ∀ l ≥ φ n, (u l, u <| φ n) ∈ V n⟩ :=
  extraction_forall_of_eventually' this

Result:
case intro.intro
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
this : ∀ (n : ℕ), ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n
φ : ℕ → ℕ
φ_extr : StrictMono φ
hφ : ∀ (n l : ℕ), l ≥ φ n → (u l, u (φ n)) ∈ V n
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n

Proof state:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n

Tactic:
have : ∀ n, ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n := fun n => by
  rw [cauchySeq_iff] at hu
  rcases hu _ (hV n) with ⟨N, H⟩
  exact ⟨N, fun k hk l hl => H _ (le_trans hk hl) _ hk⟩

Result:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
this : ∀ (n : ℕ), ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n
⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n

Proof state:
case intro
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V
n N : ℕ
H : ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V n
⊢ ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n

Tactic:
exact ⟨N, fun k hk l hl => H _ (le_trans hk hl) _ hk⟩

Result:
no goals

Proof state:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : CauchySeq u
n : ℕ
⊢ ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n

Tactic:
rw [cauchySeq_iff] at hu

Result:
α : Type u
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
hu : ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V
n : ℕ
⊢ ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n