If $A$ and $B$ are compact, then so is $A+B$.
Posting André's comment for the sake of having an answer with positive score (to prevent future bumps):
The sum is a continuous operation. The image A+B of the compact set A×B is therefore compact.
If $A$ and $B$ are compact, then so is $A+B$.
Posting André's comment for the sake of having an answer with positive score (to prevent future bumps):
The sum is a continuous operation. The image A+B of the compact set A×B is therefore compact.