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Math Help - Cantor Intersection Theorem

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    Cantor Intersection Theorem

    Suppose that H is a closed bounded set of real numbers and that (U sub n) is an expanding sequence of open sets.

    (a) Explain why the sequence of sets H \ (U sub n) is a contracting sequence of closed bounded sets.

    (b) Use the Cantor intersection theorem to deduce that if H \ (U sub n) does not equal an empty set for every n, then the intersection from n=1 to infinity, of (H \ (U sub n)) does not equal an empty set.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Slazenger3 View Post
    Suppose that H is a closed bounded set of real numbers and that U_n is an expanding sequence of open sets.

    (a) Explain why the sequence of sets H-U_nis a contracting sequence of closed bounded sets.
    Note that U_n\subseteq U_{n+1}\implies H-U_{n}\supseteq H- U_{n+1}. So, they're clearly contracting. Also, We know that H is closed and U_n open, thus H-U_n=H\cap \left(U_n\right)' is the intersection of two closed sets, thus closed. Lastly, note that H\supseteq H-U_1\supseteq H-U_2\supseteq\cdots and so \text{diam }H\geqslant \text{diam }H-U_1\geqslant \text{diam }H-U_2\geqslant\cdots

    (b) Use the Cantor intersection theorem to deduce that if H-U_n does not equal an empty set for every n, \bigcap_{n=1}^{\infty}\left\{H-U_n\right\}\ne\varnothing
    Which Cantor intersection theorem? The one that deals with complete spaces or the one that deals with compact spaces? Noting the insistence on bounded subspaces I assume the latter.

    Merely note from the above that given a finite subclass H-U_{n_1},\cdots,H-U_{n_j} of \left\{H-U_n\right\}_{n\in\mathbb{N}} that \left(H-U_{n_1}\right)\cap\cdots\cap \left(H-U_{n_j}\right)=H-U_\alpha where \alpha=\max\{n_1,\cdots,n_j\} and since by assumption this is non-empty we see that \left\{H-U_n\right\}_{n\in\mathbb{N}} is a class of closed subsets of the compact space H with the FIP. It follows that \bigcap_{n=1}^{\infty}\left\{H-U_n\right\}\ne\varnothing
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