/-- This is really an auxiliary result en route to `blimsup_cthickening_ae_le_of_eventually_mul_le`
(which is itself an auxiliary result en route to `blimsup_cthickening_mul_ae_eq`).

NB: The `: Set α` type ascription is present because of
https://github.com/leanprover-community/mathlib/issues/16932. -/
theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
    (hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
    {M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
    (blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
      (blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α) := by
  /- Sketch of proof:

  Assume that `p` is identically true for simplicity. Let `Y₁ i = cthickening (r₁ i) (s i)`, define
  `Y₂` similarly except using `r₂`, and let `(Z i) = ⋃_{j ≥ i} (Y₂ j)`. Our goal is equivalent to
  showing that `μ ((limsup Y₁) \ (Z i)) = 0` for all `i`.

  Assume for contradiction that `μ ((limsup Y₁) \ (Z i)) ≠ 0` for some `i` and let
  `W = (limsup Y₁) \ (Z i)`. Apply Lebesgue's density theorem to obtain a point `d` in `W` of
  density `1`. Since `d ∈ limsup Y₁`, there is a subsequence of `j ↦ Y₁ j`, indexed by
  `f 0 < f 1 < ...`, such that `d ∈ Y₁ (f j)` for all `j`. For each `j`, we may thus choose
  `w j ∈ s (f j)` such that `d ∈ B j`, where `B j = closedBall (w j) (r₁ (f j))`. Note that
  since `d` has density one, `μ (W ∩ (B j)) / μ (B j) → 1`.

  We obtain our contradiction by showing that there exists `η < 1` such that
  `μ (W ∩ (B j)) / μ (B j) ≤ η` for sufficiently large `j`. In fact we claim that `η = 1 - C⁻¹`
  is such a value where `C` is the scaling constant of `M⁻¹` for the uniformly locally doubling
  measure `μ`.

  To prove the claim, let `b j = closedBall (w j) (M * r₁ (f j))` and for given `j` consider the
  sets `b j` and `W ∩ (B j)`. These are both subsets of `B j` and are disjoint for large enough `j`
  since `M * r₁ j ≤ r₂ j` and thus `b j ⊆ Z i ⊆ Wᶜ`. We thus have:
  `μ (b j) + μ (W ∩ (B j)) ≤ μ (B j)`. Combining this with `μ (B j) ≤ C * μ (b j)` we obtain
  the required inequality. -/
  set Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
  set Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
  let Z : ℕ → Set α := fun i => ⋃ (j) (_ : p j ∧ i ≤ j), Y₂ j
  suffices ∀ i, μ (atTop.blimsup Y₁ p \ Z i) = 0 by
    rwa [ae_le_set, @blimsup_eq_iInf_biSup_of_nat _ _ _ Y₂, iInf_eq_iInter, diff_iInter,
      measure_iUnion_null_iff]
  intros i
  set W := atTop.blimsup Y₁ p \ Z i
  by_contra contra
  obtain ⟨d, hd, hd'⟩ : ∃ d, d ∈ W ∧ ∀ {ι : Type _} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) → (∀ᶠ j in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
    Measure.exists_mem_of_measure_ne_zero_of_ae contra
      (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ W 2)
  replace hd : d ∈ blimsup Y₁ atTop p := ((mem_diff _).mp hd).1
  obtain ⟨f : ℕ → ℕ, hf⟩ := exists_forall_mem_of_hasBasis_mem_blimsup' atTop_basis hd
  simp only [forall_and] at hf
  obtain ⟨hf₀ : ∀ j, d ∈ cthickening (r₁ (f j)) (s (f j)), hf₁, hf₂ : ∀ j, j ≤ f j⟩ := hf
  have hf₃ : Tendsto f atTop atTop :=
    tendsto_atTop_atTop.mpr fun j => ⟨f j, fun i hi => (hf₂ j).trans (hi.trans <| hf₂ i)⟩
  replace hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0) := hr.comp hf₃
  replace hMr : ∀ᶠ j in atTop, M * r₁ (f j) ≤ r₂ (f j) := hf₃.eventually hMr
  replace hf₀ : ∀ j, ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
  · intro j
    specialize hrp (f j)
    rw [Pi.zero_apply] at hrp
    rcases eq_or_lt_of_le hrp with (hr0 | hrp')
    · specialize hf₀ j
      rw [← hr0, cthickening_zero, (hs (f j)).closure_eq] at hf₀
      exact ⟨d, hf₀, by simp [← hr0]⟩
    · simpa using mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))
        (by positivity) (lt_two_mul_self hrp') (hf₀ j))
  choose w hw hw' using hf₀
  let C := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
  have hC : 0 < C :=
    lt_of_lt_of_le zero_lt_one (IsUnifLocDoublingMeasure.one_le_scalingConstantOf μ M⁻¹)
  suffices ∃ η < (1 : ℝ≥0),
      ∀ᶠ j in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ η by
    obtain ⟨η, hη, hη'⟩ := this
    replace hη' : 1 ≤ η := by
      simpa only [ENNReal.one_le_coe_iff] using
        le_of_tendsto (hd' w (fun j => r₁ (f j)) hr <| eventually_of_forall hw') hη'
    exact (lt_self_iff_false _).mp (lt_of_lt_of_le hη hη')
  refine' ⟨1 - C⁻¹, tsub_lt_self zero_lt_one (inv_pos.mpr hC), _⟩
  replace hC : C ≠ 0 := ne_of_gt hC
  let b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
  let B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
  have h₁ : ∀ j, b j ⊆ B j := fun j =>
    closedBall_subset_closedBall (mul_le_of_le_one_left (hrp (f j)) hM'.le)
  have h₂ : ∀ j, W ∩ B j ⊆ B j := fun j => inter_subset_right W (B j)
  have h₃ : ∀ᶠ j in atTop, Disjoint (b j) (W ∩ B j) := by
    apply hMr.mp
    rw [eventually_atTop]
    refine'
      ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) _⟩
    change Disjoint (b j) (Z i)ᶜ
    rw [disjoint_compl_right_iff_subset]
    refine' (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans
      ((cthickening_mono hj' _).trans fun a ha => _)
    simp only [mem_iUnion, exists_prop]
    exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩
  have h₄ : ∀ᶠ j in atTop, μ (B j) ≤ C * μ (b j) :=
    (hr.eventually (IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'
      μ M hM)).mono fun j hj => hj (w j)
  refine' (h₃.and h₄).mono fun j hj₀ => _
  change μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
  rcases eq_or_ne (μ (B j)) ∞ with (hB | hB); · simp [hB]
  

Proof state:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬↑↑μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => ↑↑μ (W ∩ closedBall (w j) (δ j)) / ↑↑μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
j : ℕ
hj₀ : Disjoint (b j) (W ∩ B j) ∧ ↑↑μ (B j) ≤ ↑C * ↑↑μ (b j)
hB : ↑↑μ (B j) ≠ ⊤
⊢ ↑↑μ (W ∩ B j) / ↑↑μ (B j) ≤ ↑(1 - C⁻¹)

Next tactic:
apply ENNReal.div_le_of_le_mul