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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import Mathlib
namespace OAI
open Set Metric
open scoped BigOperators
noncomputable section
namespace CompactBanach
theorem compact_shrinking_union {B : Type*} [MetricSpace B]
(o : B) (C : ℕ → Finset B)
(hsmall : ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ j ≥ N, ∀ z ∈ C j, dist z o < ε) :
IsCompact ({o} ∪ ⋃ j : ℕ, (C j : Set B)) := by
classical
apply isCompact_of_finite_subcover
intro ι U hU hcov
obtain ⟨i₀, hi₀⟩ := mem_iUnion.mp (hcov (by simp : o ∈ {o} ∪ ⋃ j : ℕ, (C j : Set B)))
obtain ⟨ε, hε, hεU⟩ := Metric.isOpen_iff.mp (hU i₀) o hi₀
obtain ⟨N, hN⟩ := hsmall ε hε
let F : Finset B := (Finset.range N).biUnion C
have hFK : (F : Set B) ⊆ {o} ∪ ⋃ j : ℕ, (C j : Set B) := by
intro z hz
obtain ⟨j, _, hj⟩ := Finset.mem_biUnion.mp hz
exact Or.inr (mem_iUnion.mpr ⟨j, hj⟩)
obtain ⟨t, ht⟩ := F.finite_toSet.isCompact.elim_finite_subcover U hU (hFK.trans hcov)
refine ⟨insert i₀ t, ?_⟩
intro z hz
rcases hz with hz | hz
· rcases Set.mem_singleton_iff.mp hz with rfl
exact mem_iUnion.mpr ⟨i₀, mem_iUnion.mpr ⟨by simp, hi₀⟩⟩
· obtain ⟨j, hj⟩ := mem_iUnion.mp hz
by_cases hjN : j < N
· have hzF : z ∈ F := Finset.mem_biUnion.mpr ⟨j, Finset.mem_range.mpr hjN, hj⟩
obtain ⟨i, hi⟩ := mem_iUnion.mp (ht hzF)
obtain ⟨hit, hzi⟩ := mem_iUnion.mp hi
exact mem_iUnion.mpr ⟨i, mem_iUnion.mpr ⟨Finset.mem_insert_of_mem hit, hzi⟩⟩
· have hzi : z ∈ U i₀ := hεU (hN j (by omega) z hj)
exact mem_iUnion.mpr ⟨i₀, mem_iUnion.mpr ⟨by simp, hzi⟩⟩
end CompactBanach
end
end OAI
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