Proposition 4

Tag 03JK, permanent. Cite it as [Bourbaki, Tag 03JK].

Theory of Sets, chapter III, § 1. Order relations. Ordered sets

Let E be an ordered set and let A be a subset of E which has both a greatest lower bound and a least upper bound in E. Then $\inf A \leqslant \sup A$ if $A \neq \emptyset$ ; if $A = \emptyset$ , $\sup A$ is the least and $\inf A$ the greatest element of E.

This follows immediately from the definitions.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

top-iv-s4-prop-4 not in this corpus

Cites

Nothing.