Let $M$ be a metric space and suppose that $K \subset M$ is a non-empty compact set. So if $p$ is any element of $M$, then there is a point $q$ that belongs to $K$, such that $d(p,x)\leq d(p,q)$ for every $x$ that belongs to $K$. Prove this using the the definition of compactness in terms of open coverings, and again, using the limit point property.
I got the first part, any help with the second?