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

If $\displaystyle {(x_{n},y_{n})}$ is a sequence of points in they $\displaystyle xy$ plane, all of which lie in a rectangle,

$\displaystyle R = [a,b] \times [c,d] = {(x,y): a \leq x \leq b, c \leq y \leq d},$

then there is a subsequence $\displaystyle {(x_{n_{i}},y_{n_{i}})}$ which converges (i.e., the $\displaystyle x's$ and $\displaystyle y's$ each form a convergent sequence)