Sum of two closed sets in $\mathbb R$ is closed? Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it?
(By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in $Y$)
Thanks!

consider $\mathbb Z$ and $\sqrt 2 \mathbb Z$ both are closed but the sum is not...:)
moreover it is dense on $\mathbb R$
