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Math Help - Bolzano-Weierstrass Theorem

  1. #1
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    Bolzano-Weierstrass Theorem

    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)
    Last edited by cgiulz; September 24th 2009 at 04:23 PM.
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  2. #2
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    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.
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  3. #3
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    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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