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Math Help - Could someone help me prove this.

  1. #1
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    Could someone help me prove this.

    Here's the problem

    Show that S, subset R, is compact iff every infinite subset of S contains at least one accumulation point.

    I've proven the => part:

    Suppose that S was compact. THen, let T be an infinite subset of S. Then, since every subset of a compact set is compact, T is compact. Hence, T is bounded. Thus by the Bolzano-Weierstrass theorem, T has an accumulation point.

    But, I'm having a good deal of difficulty proving the converse.
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  2. #2
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    every subset of a compact set is compact
    This is not true, but I'll leave the counterexample to you (it's not difficult).

    Quote Originally Posted by Chris11 View Post
    Show that S, subset R, is compact iff every infinite subset of S contains at least one accumulation point.
    For \Rightarrow the idea is to assume there exists a sequence T=(x_k)_k with no accumulation points in S, then there exist intervals U_{k} with center at x_k that contain (from T) only x_k (why?). Clearly T is closed (why?) so it is compact (why?), but can the cover \{ U_k \} have a finite subcover? (I picked a sequence because it's easier to visualize, but T need not be countable)

    For \Leftarrow just use Heine Borel.
    Last edited by Jose27; July 8th 2010 at 06:25 PM. Reason: Wrong theorem
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