How to show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$? I really think I have no talents in topology. This is a part of a problem from Topology by Munkres:

Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$. 

I always have the feeling that it is easy to understand the problem emotionally but hard to express it in math language. I am a student in Economics and I DO LOVE MATH. I really want to learn math well, could anyone give me some advice. Thanks so much!

Hint: Compact $\iff$ sequentially compact for metric spaces. Can you construct a sequence which must tend to the $a$ you want to find?
