theorem extracted_formal_statement_327.{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] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : (Fin n → Ω) × (Fin n → Ω)) (h_original_goal : ‖⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|‖ ≤ ↑n * (b + b))
  (i : ι) : |(|∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)| ≤ ↑n * (b + b) := sorry

State before:
case h
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ ‖⨆ i, |∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|‖ ≤ ↑n * (b + b)

Tactic: apply abs_iSup_le

State after:
[case h.hf
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 X : Ω → Z
 inst_2 : Nonempty ι
 inst_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 ⊢ ∀ (i : ι), |(|∑ k, (f i (X (ω.1 k)) - f i (X (ω.2 k)))|)| ≤ ↑n * (b + b)]

Relevant lemmas:
abs_iSup_le : ∀ {ι : Type v} [inst : Nonempty ι] {f : ι → ℝ} {a : ℝ}, (∀ (i : ι), |f i| ≤ a) → |⨆ i, f i| ≤ a