Proposition 6

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

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

Let $f$ and $g$ be two real-valued functions defined on $X$ . If $f(x) \leq g(x)$ at every point $x$ of a non-empty subset $A$ of $X$ , then we have

\begin{cases} \sup_{x \in A} f(x) \leq \sup_{x \in A} g(x), \\ \inf_{x \in A} f(x) \leq \inf_{x \in A} g(x). \end{cases}

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

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