# Closed sets

## Homework Statement

Given A and B are closed sets in R does it follow that A+B is closed? (A+B={a+b|a in A and b in B})

## Homework Equations

A set X is closed iff all of its limiting points are in X.

## The Attempt at a Solution

I don't think this is true. I've tried constructing convergent sequences A and B and having the limit of the sum not being contained in A+B. But then A and B can't be closed.

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Dick
Homework Helper
Ok, let A=N where N={1,2,3...}. Let B={-n+1/(n+1)} for n in N. I claim 0 is in the closure of A+B. But is not in A+B. Can you prove me wrong? Whew, that took a while.

Gah. That's nice and clean. Thanks.

Dick
Homework Helper
No problem. Though it did hurt. I figure if you are having problems, it's not going to be easy for me, either.

StatusX
Homework Helper
Incidentally, if A and B are both bounded, A+B is closed. I'm not sure about the case when only one of them is bounded though.

Dick