The bounded sequence has a convergent subsequence .
The bounded subsequence has a convergent subsubsequence .
The (sub)subsequence does the trick.
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)