theorem extracted_formal_statement_36.{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] (γ : ℝ)
  (hγ : γ > 1) (ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)) (dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l))
  (bd : Bornology.IsBounded (z '' univ)) (lbdψ : BddBelow (ψ '' univ)) (a b : Set (WithLp 2 (E × F)))
  (closea : IsClosed a) (closeb : IsClosed b) (sub_ab : limit_set z ⊆ a ∪ b) (nez_a : (limit_set z ∩ a).Nonempty)
  (nez_b : (limit_set z ∩ b).Nonempty) (nz_ab : limit_set z ∩ (a ∩ b) = ∅) :
  let A := limit_set z ∩ a;
  let B := limit_set z ∩ b;
  A ∪ B = limit_set z →
    A ∩ B = ∅ →
      let ω := fun z => (EMetric.infEdist z A).toReal / ((EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal);
      (∀ (z : WithLp 2 (E × F)), (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal ≠ 0) →
        Continuous ω →
          let U := {z | ω z < 1 / 4};
          let V := {z | 3 / 4 < ω z};
          (∀ z_ ∈ A, ω z_ = 0) →
            (∀ z_ ∈ B, ω z_ = 1) →
              ∀ (h : ∀ (k0 : ℕ), ∃ k, k0 ≤ k ∧ z k ∉ U ∧ z k ∉ V),
                let W := fun n => Nat.recOn n (Classical.choose (h 0)) fun n p => Classical.choose (h (p + 1));
                StrictMono W →
                  Bornology.IsBounded (z ∘ W '' univ) →
                    ∀ 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
γ : ℝ
hγ : γ > 1
ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)
dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)
bd : Bornology.IsBounded (z '' univ)
lbdψ : BddBelow (ψ '' univ)
a b : Set (WithLp 2 (E × F))
closea : IsClosed a
closeb : IsClosed b
sub_ab : limit_set z ⊆ a ∪ b
nez_a : (limit_set z ∩ a).Nonempty
nez_b : (limit_set z ∩ b).Nonempty
nz_ab : limit_set z ∩ (a ∩ b) = ∅
A : Set (WithLp 2 (E × F)) := limit_set z ∩ a
B : Set (WithLp 2 (E × F)) := limit_set z ∩ b
eqAB : A ∪ B = limit_set z
disjoint_AB : A ∩ B = ∅
ω : WithLp 2 (E × F) → ℝ :=
  fun z => (EMetric.infEdist z A).toReal / ((EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal)
sum_ne_zero : ∀ (z : WithLp 2 (E × F)), (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal ≠ 0
contω : Continuous ω
U : Set (WithLp 2 (E × F)) := {z | ω z < 1 / 4}
V : Set (WithLp 2 (E × F)) := {z | 3 / 4 < ω z}
A0 : ∀ z_ ∈ A, ω z_ = 0
B1 : ∀ z_ ∈ B, ω z_ = 1
h : ∀ (k0 : ℕ), ∃ k, k0 ≤ k ∧ z k ∉ U ∧ z k ∉ V
W : ℕ → ℕ := fun n => Nat.recOn n (Classical.choose ⋯) fun n p => Classical.choose ⋯
monoW : StrictMono W
bd' : Bornology.IsBounded (z ∘ W '' univ)
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