Bourbaki

Exercise 5, § 5

General Topology, IV, 390

Let f,gf, g be two real-valued functions defined on a filtered set. Show that if lim⁡g\lim g exists and is ⩾0\geqslant 0, then

lim⁡sup⁡fg=(lim⁡sup⁡f)(lim⁡g) \lim \sup fg = (\lim \sup f)(\lim g)
whenever both sides are defined.

Solution

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Transcribed from the printed volume. Not translated.