Corollary 1

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

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

Let $f$ and $g$ be two real-valued functions defined on a filtered set $X$ . If $\lim g$ exists, then

(33) \quad \lim \sup (f + g) = \lim \sup f + \lim g

whenever both sides are defined, and

(34)

\lim \sup fg = (\lim \sup f) (\lim g)

whenever both sides are defined and $f$ and $g$ are $\geqslant 0$ .

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

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