theorem extracted_formal_statement_309.{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)
  (this :
    (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
      ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ)
  (h : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b))
  (h_original_goal :
    ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
      ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) :
  ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
    ∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ := sorry

State before:
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
this :
  (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
    ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ
h : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)
⊢ ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
    ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ

Tactic: apply le_trans _ this

State after:
[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
 this :
   (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
     ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ
 h : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)
 ⊢ ‖∫ (x : Fin n → Ω), ⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))| ∂Measure.pi fun x => μ‖ ≤
     ∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ]

Relevant lemmas:
le_trans : ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b ≤ c → a ≤ c