-- Module: Mathlib.MeasureTheory.OuterMeasure.Basic
Goal:
α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : Ne (α := ℝ≥0∞) (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + 1) \ s k)) ∞
refine : Tendsto (β := ℝ≥0∞) (fun (k : ℕ) ↦ (m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s k)) atTop (𝓝 0)
refine₁ :
  ∀ (n : ℕ),
    LE.le (α := (fun (a : ℕ) ↦ ℝ≥0∞) n) ((m : (a : Set α) → ℝ≥0∞) ((⋃ (n : ℕ), s n) \ s n) : ℝ≥0∞)
      (∑' (k : ℕ), (m : (a : Set α) → ℝ≥0∞) (s (k + n + 1) \ s (k + n)) : ℝ≥0∞)
refine₂ : ∀ (n : ℕ), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₃ : ∀ (n : ℕ) (h' : Monotone s), (⋃ (n : ℕ), s n) \ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₄ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₅ : ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₆ :
  ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i),
    x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₇ :
  ∀ (n : ℕ) (h' : Monotone s) (i : ℕ) ⦃x : α⦄ (hx : x ∈ s i) (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j)
    (hlt : ∀ m < j, x ∉ s m), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
refine₈ :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m),
    x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
inl :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m)
    (hjn : j ≤ n), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
inr inr₁ :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m)
    (hnj : n < j), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
inr₂ :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m)
    (hnj : n < j) (this : j - (n + 1) + n + 1 = j), x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \ s (i + n)
inr₃ :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m)
    (hnj : n < j) (this : j - (n + 1) + n + 1 = j), x ∈ s (j - (n + 1) + n + 1)
inr₄ inr₅ :
  ∀ (n : ℕ) (h' : Monotone s) ⦃x : α⦄ (this : ∃ (i : ℕ), x ∈ s i) (j : ℕ) (hj : x ∈ s j) (hlt : ∀ m < j, x ∉ s m)
    (hnj : n < j) (this : j - (n + 1) + n + 1 = j), j - (n + 1) + n < j
⊢ Eq (α := ℝ≥0∞) ((m : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), s n) : ℝ≥0∞) (⨆ (n : ℕ), (m : (a : Set α) → ℝ≥0∞) (s n) : ℝ≥0∞)

Tactic:
classical
  refine measure_iUnion_of_tendsto_zero m atTop ?_
  refine tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => ?_
  refine (m.mono ?_).trans (measure_iUnion_le _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃ i, x ∈ s i := by exists i
  rcases Nat.findX this with ⟨j, hj, hlt⟩
  clear hx i
  rcases le_or_gt j n with hjn | hnj
  · exact Or.inl (h' hjn hj)
  have : j - (n + 1) + n + 1 = j := by lia
  refine Or.inr (mem_iUnion.2 ⟨j - (n + 1), ?_, hlt _ ?_⟩)
  · rwa [this]
  · rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]

Premises used: Nat.succ_eq_add_one, Set.iUnion_subset_iff, Set.diff_subset_iff, MeasureTheory.OuterMeasure.mono, tendsto_nhds_bot_mono', MeasureTheory.measure_iUnion_of_tendsto_zero, Filter.atTop, ENNReal.tendsto_sum_nat_add, LE.le.trans, Nat.succ_le_iff