theorem extracted_formal_statement_134.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] {f : E → ℝ} {g : F → ℝ} {H : WithLp 2 (E × F) → ℝ}
  {x0 : E} {y0 : F} {l : NNReal} {alg : BCD f g H l x0 y0} [inst_6 : ProperSpace E] [inst_7 : ProperSpace F]
  (bd : Bornology.IsBounded (z '' univ)) (ε : ℝ) (epos : 0 < ε)
  (h :
    ∀ (x : ℕ),
      ∃ x_1,
        x ≤ x_1 ∧
          (-ε < (EMetric.infEdist (z x_1) (limit_set z)).toReal →
            ε ≤ (EMetric.infEdist (z x_1) (limit_set z)).toReal)) :
  let W := fun n => Nat.recOn n (Classical.choose (h 0)) fun n p => Classical.choose (h (p + 1));
  StrictMono W →
    IsSeqCompact (closure (z ∘ W '' univ)) →
      (∀ (n : ℕ), (z ∘ W) n ∈ closure (z ∘ W '' univ)) →
        ∃ z_ ∈ closure (z ∘ W '' univ), ∃ α, StrictMono α ∧ Tendsto (fun n => (z ∘ W) (α n)) atTop (𝓝 z_) := sorry

State before:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : CompleteSpace F
f : E → ℝ
g : F → ℝ
H : WithLp 2 (E × F) → ℝ
x0 : E
y0 : F
l : NNReal
alg : BCD f g H l x0 y0
inst✝¹ : ProperSpace E
inst✝ : ProperSpace F
bd : Bornology.IsBounded (z '' univ)
ε : ℝ
epos : 0 < ε
h :
  ∀ (x : ℕ),
    ∃ x_1,
      x ≤ x_1 ∧
        (-ε < (EMetric.infEdist (z x_1) (limit_set z)).toReal → ε ≤ (EMetric.infEdist (z x_1) (limit_set z)).toReal)
W : ℕ → ℕ := fun n => Nat.recOn n (Classical.choose ⋯) fun n p => Classical.choose ⋯
monoW : StrictMono W
hcs : IsSeqCompact (closure (z ∘ W '' univ))
even : ∀ (n : ℕ), (z ∘ W) n ∈ closure (z ∘ W '' univ)
⊢ ∃ z_ ∈ closure (z ∘ W '' univ), ∃ α, StrictMono α ∧ Tendsto (fun n => (z ∘ W) (α n)) atTop (𝓝 z_)

Tactic: apply hcs.subseq_of_frequently_in (Filter.Frequently.of_forall even)

State after:

Relevant lemmas:
Filter.Frequently.of_forall : ∀ {α : Type u} {f : Filter α} [inst : f.NeBot] {p : α → Prop},
  (∀ (x : α), p x) → ∃ᶠ (x : α) in f, p x