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Math Help - Union of infinite number of sets is not empty

  1. #1
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    Union of infinite number of sets is not empty

    Let [], be closed intervals with [] [] for all . Prove that

    I know it has something to do with compactness, but I am not sure how to proceed.
    Last edited by Pythagorean12; December 1st 2009 at 03:07 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pythagorean12 View Post
    Let [], be closed intervals with [] [] for all . Prove that

    I know it has something to do with compactness, but I am not sure how to proceed.
    What have you tried?!
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  3. #3
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    The requirement that \left( {\forall m,n} \right)\left[ {[a_n ,b_n ] \cap [a_m ,b_m ] \ne \emptyset } \right] implies that \left( {\forall m,n} \right)\left[ {a_n  \leqslant b_m } \right].

    So A = \left\{ {a_n :n \in \mathbb{Z}^ +  } \right\} has a least upper bound. Call it \alpha.

    Show that \alpha  \in \bigcap\limits_n {\left[ {a_n ,b_n } \right]} .
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  4. #4
    Senior Member Shanks's Avatar
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    The statement implies that The bounded colsed interval collection has finite intersection property,thus the intersection is nonempty.
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