Closed Sets A & B: Does A+B Follow Suit?

[ZioX]

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.

 

[Dick]

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.

 

[ZioX]

Gah. That's nice and clean. Thanks.

 

[Dick]

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

 

[StatusX]

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]

Well, if the C sequence is convergent and A is bounded then the A sequence has a convergent subsequence (compact). Doesn't that imply the corresponding B subsequence is convergent and seal everyone's fate?

 

[StatusX]

Not bad. My idea for the case where both were compact was to send AxB under +:RxR->R, the product and images of compact sets being compact.