If A and B are compact than also A+B Suppose we have a topological vector space $X$ and $A, B\subset X$. We define A+B to be the set of the sums $a+b$ where $a\in A$ and $b\in B$. We should prove that also A+B is compact if A and B are compact. But the union of arbitrary compact sets isn't compact in generell. Thus: why is the statement wel true? 

The function $F(x,y): X \times X \to X$ defined by $F(x,y)=x+y$ is continuous, and $A \times B$ is compact in $X \times Y$. 
The image of a compact set under an continuous function is compact.
