theorem extracted_formal_statement_139.{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_) → map z (map W atTop) ≤ map (fun n => z n) atTop := 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_)
⊢ map z (map W atTop) ≤ map (fun n => z n) atTop

Tactic: apply map_mono (StrictMono.tendsto_atTop monoW)

State after:

Relevant lemmas:
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