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)

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- September 24th 2009, 03:05 PMcgiulzBolzano-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) - September 25th 2009, 01:30 PMhalbard
The bounded sequence has a convergent subsequence .

The bounded subsequence has a convergent subsubsequence .

The (sub)subsequence does the trick. - September 29th 2009, 12:10 PMcgiulz
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.