Proposition 4

Tag 0B5G, permanent. Cite it as [Bourbaki, Tag 0B5G].

General Topology, chapter I, § 7. Limits

A point x is a cluster point of a filter $\mathfrak{F}$ if and only if there is a filter finer than $\mathfrak{F}$ which converges to x.

In particular, every limit point of a filter $\mathfrak{F}$ is a cluster point of $\mathfrak{F}$ .

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

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Nothing.