Bolzano-Weierstrass Theorem

Prove the following two dimensional form of the Bolzano-Weierstrass Theorem:

If is a sequence of points in they plane, all of which lie in a rectangle,



then there is a subsequence which converges (i.e., the and each form a convergent sequence)

The bounded sequence has a convergent subsequence .

The bounded subsequence has a convergent subsubsequence .

The (sub)subsequence does the trick.

Thanks for your very helpful reply.

I'm a bit slow with this theoretical business, how do we know is bounded and is not?

Thank you very much.