1. The problem statement, all variables and given/known data I am trying to prove that, if X is compact and Y is closed, X+Y is closed. Both X and Y are sets of real numbers. 2. Relevant equations 3. The attempt at a solution I know that a sum of two closed sets isn't necessarily closed. So I presume the key must be the difference between the compact set and the closed set, namely the boundedness of a compact set. But I'm not sure how to exploit that fact for a proof.