TITLE: Compactness used to get a covering by special smaller balls
QUESTION [5 upvotes]: Suppose $(X,d)$ is a compact metric space. Suppose we have a set $A \subseteq X$ such that the set of open $\epsilon$-balls around the points of $A$ cover $X$. I've read that "By compactness, there exists $\epsilon_{1} < \epsilon$ such that the $\epsilon_{1}$-balls centered at the points of $A$ also cover $X$." Do any of you have hints for how to think about this/get running with this? I know that compact metric spaces are totally bounded, so for any $\epsilon_{2}$ there's a finite collection of $\epsilon_{2}$-balls that cover $X$, but then we don't have any information about whether or not (any) of those balls are centered around our desired points. 
There's a little bit of information about $A$ that I don't think is needed - if it is, however, I'll gladly provide it. 
Thanks for any hints!

REPLY [6 votes]: Consider the open cover $B(a, \delta)$ for all $a \in A$ and all $\delta < \epsilon$. Since $X$ is compact, there is a finite subcover $B(a_i, \delta_i)$ for $i \in \{1, \ldots, n\}$. Now put $\epsilon_1 = \max(\delta_i : i \in \{1, \ldots, n\})$.