Corollary 1

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

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

In order that $\lim \sup_{\mathfrak{G}} f = \lim \inf_{\mathfrak{G}} f$ , it is necessary and sufficient that $f$ has a limit with respect to the filter $\mathfrak{G}$ , and then

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

For since

\overline{\mathbf{R}}

is compact, the filter base

f(\mathfrak{G})

has a limit point if and only if it has only one cluster point (Chapter I, § 9, no. 1, Corollary to Theorem 1).

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

Cited by

Proposition 11, § 5

Cites

Chapter I, § 9, no. 1, Corollary to Theorem 1