Corollary 1

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

General Topology, chapter IV, § 5. Real-valued functions

Let $f$ be a real-valued function defined on $X$ , and let $k$ be a real number. Then (22)

\sup_{x \in A} (f(x) + k) = k + \sup_{x \in A} f(x)

whenever both sides are defined, and, if $k \geq 0,$ (23)

\sup_{x \in A} (kf(x)) = k \cdot \sup_{x \in A} f(x)

whenever both sides are defined.

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

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