theorem extracted_formal_statement_169.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha'' :
    ∀ (s : Set ℝ) (x : ℝ),
      (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s)
  (s : Set ℝ)
  (ha' :
    ∀ (s x : Set ℝ),
      x ⊆ s →
        (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) →
          0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s)
  (t : Set ℝ) (hs₁ : t ⊆ s) (hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t) (hs₀ : 0 ∈ t) (ε : ℝ) (εpos : 0 < ε)
  (hs₂₀ : Metric.ball 0 ε ⊆ t)
  (ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2)
  (hG : ¬subgradient_method.G f x₀ = 0) (hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0) (x₁ : ℝ)
  (hx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)) (x₂ : ℝ)
  (hx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2)
  (h_original_goal : |x₂| ≤ |x₁| + dist x₂ x₁) : dist x₂ 0 ≤ dist x₁ 0 + dist x₁ x₂ := sorry

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha'' :
  ∀ (s : Set ℝ) (x : ℝ),
    (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
s✝ : Set ℝ
ha' :
  ∀ (s x : Set ℝ),
    x ⊆ s →
      (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
t : Set ℝ
hs₁ : t ⊆ s✝
hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
hs₀ : 0 ∈ t
ε : ℝ
εpos : 0 < ε
hs₂₀ : Metric.ball 0 ε ⊆ t
ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
hG : ¬subgradient_method.G f x₀ = 0
hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
c₁ : ∀ x_1 ∈ s, x_1 ∈ s
x₁ : ℝ
hx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
x₂ : ℝ
hx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2
⊢ |x₂| ≤ |x₁| + dist x₂ x₁

Tactic: rw [← Real.dist_0_eq_abs x₁, ← Real.dist_0_eq_abs x₂, dist_comm x₂ x₁]

State after:
[E : Type u_1
 inst✝¹ : NormedAddCommGroup E
 inst✝ : InnerProductSpace ℝ E
 f : E → ℝ
 xm x₀ : E
 alg : subgradient_method f x₀
 hm : IsMinOn f Set.univ xm
 ha'' :
   ∀ (s : Set ℝ) (x : ℝ),
     (∀ (b : ℝ), x ≤ b → b ∈ s) → ∃ a, ∀ (b : ℕ), a ≤ b → (Finset.range (b + 1)).sum (subgradient_method.a f x₀) ∈ s
 s✝ : Set ℝ
 ha' :
   ∀ (s x : Set ℝ),
     x ⊆ s →
       (∀ x_1 ∈ x, ∃ ε, 0 < ε ∧ Metric.ball x_1 ε ⊆ x) → 0 ∈ x → ∃ a, ∀ (b : ℕ), a ≤ b → subgradient_method.a f x₀ b ∈ s
 t : Set ℝ
 hs₁ : t ⊆ s✝
 hs₂ : ∀ x ∈ t, ∃ ε, 0 < ε ∧ Metric.ball x ε ⊆ t
 hs₀ : 0 ∈ t
 ε : ℝ
 εpos : 0 < ε
 hs₂₀ : Metric.ball 0 ε ⊆ t
 ha₁ : ∃ a₁, ∀ (b : ℕ), a₁ ≤ b → ‖x₀ - xm‖ ^ 2 / (2 * (Finset.range (b + 1)).sum (subgradient_method.a f x₀)) < ε / 2
 hG : ¬subgradient_method.G f x₀ = 0
 hpos' : ↑(subgradient_method.G f x₀) ^ 2 > 0
 s : Set ℝ := {x | |x| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)}
 c₁ : ∀ x_1 ∈ s, x_1 ∈ s
 x₁ : ℝ
 hx₁ : |x₁| < ε / (2 * ↑(subgradient_method.G f x₀) ^ 2)
 x₂ : ℝ
 hx₂ : dist x₂ x₁ < (ε / (2 * ↑(subgradient_method.G f x₀) ^ 2) - |x₁|) / 2
 ⊢ dist x₂ 0 ≤ dist x₁ 0 + dist x₁ x₂]

Relevant lemmas:
dist_comm : ∀ {α : Type u} [inst : PseudoMetricSpace α] (x y : α), dist x y = dist y x
Real.dist_0_eq_abs : ∀ (x : ℝ), dist x 0 = |x|