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!

It's worth mentioning that :
if one is closed + bounded, another one is closed,then the addition is closed
Since closedness can be charaterized by sequence in $\Bbb{R}^n$,if $(x_n) \in A+B$  we need to show limit of the convergence sequence still lies in it.assume $A$ is compact $B $ is closed.
Since $x_n= a_n +b_n \to x$,compactness implies sequential compactness hence $a_{n_k} \to a\in A$  for some subsequence. now $x_{n_k} \to x$ which means subsequence $b_{n_k}\to x-a$ converge,since $B$ is closed,$x-a \in B$ ,hence $x = a+b \in A+B$,which means the sum is closed.
