Exercise 3, § 5

Let $f$ be a real-valued function defined on an infinite set $X$ and let $\mathfrak{G}$ be the filter of complements of finite subsets of $X$. Show that $\lim \sup_{\mathfrak{G}} f$ is the least upper bound of the set of real numbers $x$ such that the set $f([x, +\infty])$ is infinite.