How do I prove that a closed subset of a compact set is compact directly from the definition?

Question

I'm trying to prove that a closed subset of a compact set is compact directly from the definition but I'm not sure how to proceed.

This is what I have so far:
Let $K$ be a compact set and let $A$ be a closed subset of $K$.
As $A$ is a closed set, any convergent sequence in $A$ must converge to a limit point in $A$
Let $a_n$ be a sequence in $A$. If I prove that it has a convergent subsequence, I am done

I'm not sure how to proceed from here, a hint would be appreciated.

Answer 1

A sequence in $A$ is also a sequence in $K$ hence it has a convergent subsequence . The limit of this subsequence is in $A$ because $A$ is closed