# Bolzano-Weierstrass Theorem

• Sep 24th 2009, 03:05 PM
cgiulz
Bolzano-Weierstrass Theorem
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)
• Sep 25th 2009, 01:30 PM
halbard
The bounded sequence $\displaystyle x_n$ has a convergent subsequence $\displaystyle x_{n_i}$.

The bounded subsequence $\displaystyle y_{n_i}$ has a convergent subsubsequence $\displaystyle y_{n_{m_i}}$.

The (sub)subsequence $\displaystyle (x_{n_{m_i}},y_{n_{m_i}})$ does the trick.
• Sep 29th 2009, 12:10 PM
cgiulz
I'm a bit slow with this theoretical business, how do we know $\displaystyle x_{n}$ is bounded and $\displaystyle y_{n}$ is not?