Let (X, ||$\bullet$||) be a normed vector space. Let A $\subseteq$ X be compact and B $\subseteq$ X be closed. Prove that the set A+B is closed.

Question

A+B := {x$\in$X | there exists a$\in$A, b$\in$B s.t. a+b=x}

I tried looking for a contradiction, assuming A+B to be open, and then trying to prove that either B must be open, or that A must not be compact. I know that A being compact implies that it is closed and bounded, but since A$\cup$B might be empty, I don't know how knowing that both sets are closed is helpful? I honestly don't know where to start, so any tips would be appreciated!