Suppose that $A,B$ are in $\mathbb{R}^n$, is it true that if $B$ is compact then $\operatorname{Cl}(A+B)=\operatorname{Cl}(A)+\operatorname{Cl}(B)$? I am trying to prove that if a sequence ($a_n + b_n$) in $A+B$ converges to a+b, then both sequences $(a_n)$ and $(b_n)$ converge. I tried subsequences to conclude that $a_{n_k}$ and $b_{n_k}$ do converge, but i'm afraid i'm still missing something out.
edit: I mean $A+B=\{a+b: a \in A\text{ and }b \in B\}$, and $\operatorname{Cl}(X)$ the closure of $X$.