Proposition 4

Tag 0BQP, permanent. Cite it as [Bourbaki, Tag 0BQP].

General Topology, chapter IV, § 4. The extended real line

Every non-empty subset of $\overline{\mathbf{R}}$ has a least upper bound and a greatest lower bound.

The least upper bound (resp. greatest lower bound) of a non-empty subset $A$ of $\mathbf{R}$ is denoted by $\sup A$ (resp. $\inf A$ ). Clearly

(1)

\inf A \leq \sup A.

If $A \subset B$ , then $\sup A \leq \sup B$ and $\inf A \geq \inf B$ (Set Theory, Chapter III, § 1, no. 9, Proposition 4).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.