theorem extracted_formal_statement_27.{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 →
    ∀ z_ ∈ closure (z ∘ W '' univ),
      ∀ (α : ℕ → ℕ), StrictMono α → Tendsto (fun n => (z ∘ W) (α n)) atTop (𝓝 z_) → z_ ∈ limit_set 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
z_ : WithLp 2 (E × F)
left✝ : z_ ∈ closure (z ∘ W '' univ)
α : ℕ → ℕ
monoa : StrictMono α
conv : Tendsto (fun n => (z ∘ W) (α n)) atTop (𝓝 z_)
⊢ z_ ∈ limit_set z

Tactic: simp [limit_set, MapClusterPt]
    apply ClusterPt.mono (ClusterPt.of_le_nhds conv)
    calc
  _ = map (fun n ↦ (alg.z ∘ W) n) (map α atTop) := by rw [map_map]; rfl
  _ ≤ map (fun n ↦ (alg.z ∘ W) n) atTop := (map_mono (StrictMono.tendsto_atTop monoa))
  _ ≤ map (fun n ↦ alg.z n) atTop := by rw [← map_map]; apply map_mono (StrictMono.tendsto_atTop monoW)

State after:

Relevant lemmas:
ClusterPt.mono : ∀ {X : Type u} {x : X} [inst : TopologicalSpace X] {f g : Filter X},
  ClusterPt x f → f ≤ g → ClusterPt x g
Filter.map_map : ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : α → β} {m' : β → γ},
  map m' (map m f) = map (m' ∘ m) f
ClusterPt.of_le_nhds : ∀ {X : Type u} {x : X} [inst : TopologicalSpace X] {f : Filter X},
  f ≤ 𝓝 x → ∀ [inst_8 : f.NeBot], ClusterPt x f
Filter.map_mono : ∀ {α : Type u_1} {β : Type u_2} {m : α → β}, Monotone (map m)
StrictMono.tendsto_atTop : ∀ {φ : ℕ → ℕ}, StrictMono φ → Tendsto φ atTop atTop