Proposition 13

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

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

Let $f$ and $g$ be two real-valued functions defined on a filtered set $X$ .

(i) We have

(28) \quad \lim \sup (f + g) \leq \lim \sup f + \lim \sup g,

(29) \quad \lim \sup f + \lim \inf g \leq \lim \sup (f + g)

whenever both sides of these inequalities are defined.

(ii) If $f$ and $g$ are $\geq 0$ on $X$ , we have

(30) \quad \lim \sup fg \leq (\lim \sup f) (\lim \sup g),

(31) \quad (\lim \sup f) (\lim \inf g) \leq \lim \sup fg

whenever both sides of these inequalities are defined.

(iii) If $f \geq 0$ on $X$ , then

(32) \quad \lim \sup (1/f) = 1/(\lim \inf f)

(putting $1/0 = + \infty$ ).

These relations are consequences of Proposition 12 and relations (10)

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

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