Metric spaces and Sets...

Hello!

Here's a problem I have:

Let the distance from set A to set B be defined as:

dist(A,B) = inf {d(x,y) : x is in A, y is in B} (d(x,y) is a metric)

A & B are closed sets in R^n. A is bounded.

Prove: dist(A,B)>0 <=> A and B are disjoint sets.

Then they ask: Is the sentence still correct if A isn't bounded?

Now, I thought I had a pretty neat proof, but I didn't use the fact that A is bounded. I didn't even use the fact that A and B are closed sets!

I shall shortly detail my proof, and you guys can tell me if I'm missing something:

assume dist(A,B)>0.

let's assume (by contraction) that A and B are not disjoint, i.e., there's a z so that z is in A and in B.

So, d(z,z) is in {d(x,y) : x in A, y in B). but d(z,z) = 0 , so (needs formal proof, but), inf{d(x,y): x in A, y in B) = 0 => dist (A,B) = 0. Contraction.

The other way is pretty similar.

What am I missing? Why do I need A and B to be closed sets? Why do I need A to be bounded?

Thanks!!!!

Tomer.