Bolzano-Weierstrass Theorem

**cgiulz** · September 24th 2009, 03:05 PM

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

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

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

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

**halbard** · September 25th 2009, 01:30 PM

The bounded sequence $x_n$ has a convergent subsequence $x_{n_i}$ .

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

The (sub)subsequence $(x_{n_{m_i}},y_{n_{m_i}})$ does the trick.

**cgiulz** · September 29th 2009, 12:10 PM

Thanks for your very helpful reply.

I'm a bit slow with this theoretical business, how do we know $x_{n}$ is bounded and $y_{n}$ is not?

Thank you very much.

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