Let $(X,\varrho)$ be a metric space and $K\subset X$ compact.
Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the reciprocal is not true.
My approach. Since $K$ is compact, every open cover of $K$ contains a finite sub-cover. But every one of the open sets of the finite sub-cover can written as a union of open balls with radius $\varepsilon$.