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Sum of two closed sets in $\mathbb R$ is closed?

Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B = \{a + b : a \in A, b \in B\}$ is not closed.

This question appears on an old analysis qual I am studying. I know that both $A, B$ must be unbounded sets, because in an earlier part of the problem I have proved that $A + B$ is closed if either of the two sets are compact. The simplest unbounded and closed subset of $\mathbb{R}$ that I know is $\mathbb{Z}$. So I was starting with $A = \mathbb{Z}$, but I'm not yet able to come up with an appropriate $B$.

Hints or solutions are greatly appreciated.

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I think this a dupe. But I am unable to find the duplicate. –  user17762 Dec 30 '12 at 18:33
    
@Marvis Here is one. –  David Mitra Dec 30 '12 at 18:35
    
@DavidMitra Good to know that my memory is good. –  user17762 Dec 30 '12 at 18:41
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marked as duplicate by David Mitra, Nameless, Marvis, Jasper Loy, Thomas Andrews Dec 30 '12 at 18:48

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1 Answer

Try $\mathbb Z+\sqrt 2\mathbb Z$.

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