sSup_nonneg — Mathlib

English

If s is bounded below and above, then sInf(s) ≤ sSup(s).

Русский

Если множество ограничено снизу и сверху, то infimum не превышает supremum.

LaTeX

$$$\\text{If } BddBelow(s) \\land BddAbove(s), \\; \\operatorname{sInf}(s) \\le \\operatorname{sSup}(s)$$$

Lean4

/-- As `sSup s = 0` when `s` is a set of reals that's either empty or unbounded above,
it suffices to show that all elements of `s` are nonnegative to show that `0 ≤ sSup s`. -/
theorem sSup_nonneg (hs : ∀ x ∈ s, 0 ≤ x) : 0 ≤ sSup s :=
  by
  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
  · exact sSup_empty.ge
  · exact sSup_nonneg' ⟨x, hx, hs _ hx⟩

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