Proving Positive Distance between Disjoint Compact Sets in R^n

[nicorette]

Homework Statement

X and Y are compact sets in R^n that are disjoint. Then there must be positive distance between the elements of these sets.

Homework Equations

The Attempt at a Solution

since X and Y are compact , X X Y is compact. Then, for the distance function d(x in X, y in Y): R^n X R^n -> R, there is a maximum and a minimum.

I think this should be a sufficient proof, although I'm not really sure how exactly to show that X X Y is compact as well.

 

[Dick]

You've got the right idea, but you haven't shown the minimum can't be zero. Have you? Do you need to show XxY is compact? Or it that something you already proved?

 

[nicorette]

I don't know how to show the minimum can't be zero. So far as X x Y being compact, I just assumed, I don't know where to start for a proof

thanks a lot

 

[Dick]

Don't worry about proving XxY compact for the moment, concentrate on the first one. Suppose the distance is zero. How can that be?