Closed Subset Addition in Metric Spaces: Real Analysis Homework Help

[Mr_Physics]

Homework Statement

Let E, F be two closed and non-empty subsets of R, where R is seen as a metric space with the distance d(a,b)=|a-b| for a,b ϵ R.

Suppose E + F := { e+f |e ϵ E, f ϵ F}. Is is true that E+F has to be closed?

Homework Equations

The Attempt at a Solution

I'm not sure how to start this one.

 

[micromass]

This is not an easy one, by far!

But consider:

$$A=\{...,-4,-3,-2,-1\}$$

and

$$B=\{1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,...\}$$

Note that E+F is closed if one of E or F is compact...

 

[Dick]

This is not an easy one, by far!

But consider:

$$A=\{...,-4,-3,-2,-1\}$$

and

$$B=\{1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,...\}$$

Note that E+F is closed if one of E or F is compact...

I'll agree it's not easy. It does take some head scratching. Here's another one to think about on a different line. Take A=Z (the integers) and B=Z*sqrt(2). micromass's suggestion is really pretty clever though once you figure it out.