Theorem 1 (Principle of extension of inequalities)

Tag 0BR5, permanent. Cite it as [Bourbaki, Tag 0BR5].

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

Let $f$ and $g$ be two real-valued functions, defined on a set $X$ filtered by a filter $\mathfrak{F}$ . If $\lim_{\mathfrak{F}} f$ and $\lim_{\mathfrak{F}} g$ exist, and if $f \leq g$ , then $\lim_{\mathfrak{F}} f \leq \lim_{\mathfrak{F}} g$ .

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

Cited by

Definition 6, § 5
Remark, § 5
Proposition 12, § 5
Proposition 5, § 5
Example, § 7
Proposition 1, § 1

Cites

Nothing.