| 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 | |