Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ≥0∞
hε : 0 < ε
hεbound : ε ≤ 1
⊢ ∃ k, ∀ μ ∈ S, 1 - ε < ↑(μ (⋃ i, ⋃ (_ : i ≤ k), U i))

Tactic:
by
  have εfin : ε ≠ ∞ := ne_top_of_le_ne_top (by simp) hεbound
  lift ε to ℝ≥0 using εfin
  obtain ⟨ε, hε', rfl⟩ : ∃ (ε' : ℝ) (hε' : 0 ≤ ε'), ε = .mk ε' hε' := ⟨↑ε, ε.2, rfl⟩
  simp only [ENNReal.coe_pos, ← NNReal.coe_lt_coe, NNReal.coe_zero, coe_mk, coe_le_one_iff,
      ← NNReal.coe_le_coe, NNReal.coe_one] at hε hεbound
  by_contra! nh
  choose μ hμInS hcontradiction using nh
  obtain ⟨μlim, _, sub, hsubmono, hμconverges⟩ :=
      hcomp.isSeqCompact (fun n ↦ subset_closure <| hμInS n)
  have Measurebound n : (μlim (⋃ (i ≤ n), U i) : ℝ) ≤ 1 - ε := calc
    (μlim (⋃ (i ≤ n), U i) : ℝ)
    _ ≤ liminf (fun k ↦ (μ (sub k) (⋃ (i ≤ n), U i) : ℝ)) atTop := by
      have hopen : IsOpen (⋃ i ≤ n, U i) := isOpen_biUnion fun i a ↦ O i
      have := ProbabilityMeasure.le_liminf_measure_open_of_tendsto hμconverges hopen
      simp_rw [Function.comp_apply, ← ennreal_coeFn_eq_coeFn_toMeasure] at this
      rw [← ofNNReal_liminf] at this
      · exact mod_cast this
      use 1
      simpa [eventually_map, eventually_atTop, forall_exists_index] using fun _ x h ↦
          (h x (by simp)).trans <| ProbabilityMeasure.apply_le_one (μ (sub x)) (⋃ i ≤ n, U i)
    _ ≤ liminf (fun k ↦ (μ (sub k) (⋃ (i ≤ sub k), U i) : ℝ)) atTop := by
      apply Filter.liminf_le_liminf
      · simp only [NNReal.coe_le_coe, eventually_atTop]
        use n + 1
        intro b hypo
        refine (μ (sub b)).apply_mono
            <| Set.biUnion_mono (fun i (hi : i ≤ n) ↦ hi.trans ?_) fun _ _ ↦ le_rfl
        exact le_trans (Nat.le_add_right n 1) (le_trans hypo (StrictMono.le_apply hsubmono))
      · use 0; simp
      · use 1
        simpa [eventually_map, eventually_atTop, forall_exists_index] using
            fun _ d hyp ↦ (hyp d (by simp)).trans (by simp)
    _ ≤ 1 - ε := by
      apply Filter.liminf_le_of_le
      · use 0; simp
      simp only [eventually_atTop, forall_exists_index]
      intro b c h
      apply le_trans (h c le_rfl)
      refine (ofReal_le_ofReal_iff (by rw [sub_nonneg]; exact hεbound)).mp ?_
      rw [ofReal_coe_nnreal]
      apply le_trans (hcontradiction (sub c))
      norm_cast
  have accumulation : Tendsto (fun n ↦ μlim (⋃ i ≤ n, U i)) atTop (𝓝 (μlim (⋃ i, U i))) := by
    simp_rw [← Set.accumulate_def, ProbabilityMeasure.tendsto_measure_iUnion_accumulate]
  rw [Cov, coeFn_univ, ← NNReal.tendsto_coe] at accumulation
  have exceeds_bound : ∀ᶠ n in atTop, (1 - ε / 2 : ℝ) ≤ μlim (⋃ i ≤ n, U i) :=
      Tendsto.eventually_const_le (v := 1)
        (by simp only [sub_lt_self_iff, Nat.ofNat_pos, div_pos_iff_of_pos_right]; positivity)
        accumulation
  suffices ∀ᶠ n : ℕ in atTop, False from this.exists.choose_spec
  filter_upwards [exceeds_bound] with n hn
  linarith [hn.trans <| Measurebound n]

Goal state after: