English
A finite union of sets is totally bounded iff each set is totally bounded.
Русский
Конечная объединение множества является полностью ограниченным тогда и только тогда, когда каждая множество полностью ограничена.
LaTeX
$$$$TotallyBounded(\\bigcup_i s_i) \\iff \\forall i, TotallyBounded(s_i).$$$$
Lean4
/-- A finite indexed union is totally bounded
if and only if each set of the family is totally bounded. -/
@[simp]
theorem totallyBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set α} :
TotallyBounded (⋃ i, s i) ↔ ∀ i, TotallyBounded (s i) :=
by
refine ⟨fun h i ↦ h.subset (subset_iUnion _ _), fun h U hU ↦ ?_⟩
choose t htf ht using (h · U hU)
refine ⟨⋃ i, t i, finite_iUnion htf, ?_⟩
rw [biUnion_iUnion]
gcongr; apply ht
