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Thread: Bolzano-Weierstrass Theorem

  1. #1
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    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)
    Last edited by cgiulz; Sep 24th 2009 at 03:23 PM.
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  2. #2
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    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.
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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 $\displaystyle x_{n}$ is bounded and $\displaystyle y_{n}$ is not?

    Thank you very much.
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