Exercise 4, § 5

General Topology, IV, 390

Let $f, g$ be two real-valued functions defined on a set $X$ filtered by a filter $\mathfrak{F}$ . Show that

\lim \sup_{\mathfrak{F}} (\sup (f, g)) = \sup (\lim \sup_{\mathfrak{F}} f, \lim \sup_{\mathfrak{F}} g).

Give an example of a set

X

filtered by a filter

\mathfrak{F}

, and an infinite family

(f_i)

of real-valued functions defined on

X

, such that

\sup_l (\lim \sup_{\mathfrak{F}} f_i) < \lim \sup_{\mathfrak{F}} (\sup f_i).

Solution

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