Corollary

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

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

Let $(x_\iota)_{\iota \in I}$ be a family of elements of an ordered set $\mathrm{E}$ which has a least upper bound in $\mathrm{E}$ . If $\mathrm{J}$ is a subset of $\mathrm{I}$ such that the family $(x_\iota)_{\iota \in J}$ has a least upper bound in $\mathrm{E}$ , we have $\sup\limits_{\iota \in J} x_\iota \leqslant \sup\limits_{\iota \in I} x_\iota$ .

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

Proposition 7, § 1

Cites

Nothing.