If A and B are compact subsets of some metric space M with metric d, then I think that the cross product $\displaystyle A\times B$ is a compact subset of the metric space $\displaystyle M^2$ where the metric in this space is defined by

$\displaystyle p((x,y),(z,w))=\sqrt{(d(x,z)^2+d(y,w)^2}$

What is the best way to show that AxB is indeed compact? Is there a particularly good way to do this? I think I have a proof showing that every sequence in AxB has a convergent subsequence.

Thanks.