Corollary 2

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

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

If $\mathfrak{H}$ is a filter finer than $\mathfrak{G}$ , we have

\lim \inf_{\mathfrak{G}} f \leq \lim \inf_{\mathfrak{H}} f \leq \lim \sup_{\mathfrak{H}} f \leq \lim \sup_{\mathfrak{G}} f.

For every cluster point of

f

with respect to

\mathfrak{H}

is also a cluster point of

f

with respect to

\mathfrak{G}

(Chapter I, § 7, no. 3). In particular, if

\lim_{\mathfrak{G}} f

exists, then

\lim \inf_{\mathfrak{G}} f \leq \lim_{\mathfrak{G}} f \leq \lim \sup_{\mathfrak{G}} f.

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

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