bddBelow — Mathlib

English

A compact set is bounded below; in particular, there is some a ∈ α such that a ≤ x for all x in the set.

Русский

Компактное множество ограничено снизу; существует a ∈ α, такого что a ≤ x для всех x в множестве.

LaTeX

$$If S is compact, then ∃ a with ∀ x∈S, a ≤ x.$$

Lean4

/-- A compact set is bounded below -/
theorem bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBelow s :=
  by
  rcases s.eq_empty_or_nonempty with rfl | hne
  · exact bddBelow_empty
  · obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
    exact ⟨a, has⟩

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