theorem extracted_formal_statement_5.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (c : ℝ)
  (hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c) :
  Summable fun n => Φ (n + 1) - Φ (n + 1 + 1) := sorry

State before:
case intro
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
c : ℝ
hn : ∀ (n : ℕ), ∑ i ∈ Finset.range n, (Φ i.succ - Φ (i.succ + 1)) ≤ c
⊢ Summable fun n => Φ (n + 1) - Φ (n + 1 + 1)

Tactic: apply summable_of_sum_range_le (c := c)
    intro n
    have : n + 1 > 0 := by linarith
    apply Φ_sub_nonneg (n + 1).toPNat'
    exact hn

State after:

Relevant lemmas:
summable_of_sum_range_le : ∀ {f : ℕ → ℝ} {c : ℝ},
  (∀ (n : ℕ), 0 ≤ f n) → (∀ (n : ℕ), ∑ i ∈ Finset.range n, f i ≤ c) → Summable f