I'm working through Martin Schechter's "Principles of Functional Analysis" (2nd ed.) and a problem concerning totally bounded subsets and $\varepsilon$-nets of normed linear vector spaces on page 96 is really bugging me.
I'll begin with the relevant definitions. Throughout, $X$ is a normed linear vector space.
Definition ($\varepsilon$-net) Let $\varepsilon > 0$ be given. A set of points $W \subseteq X$ is called an $\varepsilon$-net for a set $U \subseteq X$ if for every $x \in U$ : $\exists z \in W$ such that $\|x -z \| < \varepsilon$.
Definition (totally bounded) A subset $U \subseteq X$ is called totally bounded if for every $\varepsilon > 0$ there exists a finite set of points $W \subseteq X$ which is an $\varepsilon$-net for $U$.
The following statement in the text is left as an exercise for the reader and it's driving me crazy.
"If $U$ is totally bounded, then for each $\varepsilon >0$ it has a finite $\varepsilon$-net $W \subseteq U$. "