theorem extracted_formal_statement_255.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω ω' : Fin n → Ω)
  (h_original_goal : ‖‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖‖ ≤ ↑n * (b + b)) :
  |⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|| ≤ ↑n * (b + b) := sorry

State before:
case h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω ω' : Fin n → Ω
⊢ ‖‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|‖‖ ≤ ↑n * (b + b)

Tactic: simp only [norm_eq_abs, abs_abs]

State after:
[case h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω ω' : Fin n → Ω
 ⊢ |⨆ i, |∑ k, (f i (X (ω k)) - f i (X (ω' k)))|| ≤ ↑n * (b + b)]

Relevant lemmas:
abs_abs : ∀ {α : Type u_1} [inst : Lattice α] [inst_4 : AddGroup α] [inst_5 : AddLeftMono α] [inst_6 : AddRightMono α]
  (a : α), |(|a|)| = |a|
Real.norm_eq_abs : ∀ (r : ℝ), ‖r‖ = |r|