theorem extracted_formal_statement_43.{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_1 b : Set (WithLp 2 (E × F)))
  (closea : IsClosed a_1) (closeb : IsClosed b) (sub_ab : limit_set z ⊆ a_1 ∪ b) (nez_a : (limit_set z ∩ a_1).Nonempty)
  (nez_b : (limit_set z ∩ b).Nonempty) (nz_ab : limit_set z ∩ (a_1 ∩ b) = ∅) :
  let A := limit_set z ∩ a_1;
  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) →
              ∀ (k0 : ℕ),
                (∀ (k : ℕ), k0 ≤ k → z k ∈ U ∨ z k ∈ V) →
                  UniformContinuousOn ω (closure (z '' univ)) →
                    (∀ ε > 0,
                        ∃ δ > 0,
                          ∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε) →
                      ∀ ε > 0,
                        ∀ δ > 0,
                          (∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε) →
                            ∀ (a : ℕ),
                              (∀ (b : ℕ), a ≤ b → ‖z (b + 1) - z b‖ < δ) →
                                ∀ n ≥ a, ‖ω (z (n + 1)) - ω (z n)‖ < ε → ‖z (n + 1) - z n‖ < δ := sorry

State before:
case h
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
k0 : ℕ
hk0 : ∀ (k : ℕ), k0 ≤ k → z k ∈ U ∨ z k ∈ V
unicont : UniformContinuousOn ω (closure (z '' univ))
this : ∀ ε > 0, ∃ δ > 0, ∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε
ε : ℝ
epos : ε > 0
δ : ℝ
dpos : δ > 0
ieq : ∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε
a : ℕ
ie : ∀ (b : ℕ), a ≤ b → ‖z (b + 1) - z b‖ < δ
n : ℕ
alen : n ≥ a
⊢ ‖ω (z (n + 1)) - ω (z n)‖ < ε

Tactic: apply ieq
    apply subset_closure (mem_image_of_mem z (mem_univ (n + 1)))
    apply subset_closure (mem_image_of_mem z (mem_univ n))

State after:
[case h.a
 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
 k0 : ℕ
 hk0 : ∀ (k : ℕ), k0 ≤ k → z k ∈ U ∨ z k ∈ V
 unicont : UniformContinuousOn ω (closure (z '' univ))
 this : ∀ ε > 0, ∃ δ > 0, ∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε
 ε : ℝ
 epos : ε > 0
 δ : ℝ
 dpos : δ > 0
 ieq : ∀ x1 ∈ closure (z '' univ), ∀ x2 ∈ closure (z '' univ), ‖x1 - x2‖ < δ → ‖ω x1 - ω x2‖ < ε
 a : ℕ
 ie : ∀ (b : ℕ), a ≤ b → ‖z (b + 1) - z b‖ < δ
 n : ℕ
 alen : n ≥ a
 ⊢ ‖z (n + 1) - z n‖ < δ]

Relevant lemmas:
subset_closure : ∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], s ⊆ closure s
Set.mem_image_of_mem : ∀ {α : Type u} {β : Type v} (f : α → β) {x : α} {a : Set α}, x ∈ a → f x ∈ f '' a
Set.mem_univ : ∀ {α : Type u} (x : α), x ∈ univ