Theorem 1

Tag 0B4A, permanent. Cite it as [Bourbaki, Tag 0B4A].

General Topology, chapter I, § 6. Filters

If $\mathfrak{F}$ is any filter on a set $X$ , there is an ultrafilter finer than $\mathfrak{F}$ .

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

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Nothing.