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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 $X$.

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    What do you mean by $A+B$?2011-12-14
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    Also, $Cl(X)$ means the closure of $X$, I guess.2011-12-14
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    $a_n$ and $b_n$ need not converge in general: consider $a_n = - b_n = (-1)^n$, where $A = B = \{ -1,1\}$. Note however, that $b_n$ has a convergent subsequence $b_{n_k}$ by compactness of $B$ and $a_{n_k}$ converges since $a_{n_k} = (a_{n_k} + b_{n_k}) - b_{n_k}$.2011-12-14
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    @Karatug: $A+B=\{a+b:a\in A\text{ and }b\in B\}$.2011-12-14
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    If $B$ is compact, then $Cl(B)=B$2011-12-14
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    [related](http://math.stackexchange.com/q/60452/8271)2012-08-28

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Since $B$ is compact, it’s closed, and we’re really trying to show that $\operatorname{cl}(A+B)=\operatorname{cl}A + B$. It’s clear that $\operatorname{cl}A+B\subseteq\operatorname{cl}(A+B)$, so the problem is to show that $\operatorname{cl}(A+B)\subseteq\operatorname{cl}A+B$. Suppose that $p\in\operatorname{cl}(A+B)$. Then, as you say, there are sequences $\langle a_n:n\in\mathbb{N}\rangle$ in $A$ and $\langle b_n:n\in\mathbb{N}\rangle$ in $B$ such that $\langle a_n+b_n:n\in\mathbb{N}\rangle$ converges to $p$.

Because $B$ is compact, the sequence $\langle b_n:n\in\mathbb{N}\rangle$ must have a convergent subsequence, say $\langle b_{n_k}:k\in\mathbb{N}\rangle$, and the limit, $b$, of this subsequence must be in $B$. For $k\in\mathbb{N}$ let $x_k=a_{n_k}+b$. Then

$$\lim_{k\to\infty}\|(a_{n_k}+b_{n_k})-x_k\|=\lim_{k\to\infty}\|b_{n_k}-b\|=0\;;$$

can you take it from there to show that $\langle x_k:k\in\mathbb{N}\rangle$ converges to $p$ and therefore that $\langle a_{n_k}:k\in\mathbb{N}\rangle$ converges to $p-b$, which of course must then be in $\operatorname{cl}A$?