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Math Help - Set Theory: Show A is finite or countably infinite

  1. #1
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    Set Theory: Show A is finite or countably infinite

    I've been staring at this proof for several days now, and I have no idea where to even begin

    Let A be a set of positive real numbers such that
    Sigma: x <= 1
    ^any finite subset of A

    Show A is finite or countably infinite

    Any help is much appreciated
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  2. #2
    Grand Panjandrum
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    Re: Set Theory: Show A is finite or countably infinite

    Quote Originally Posted by jsmith90210 View Post
    I've been staring at this proof for several days now, and I have no idea where to even begin

    Let A be a set of positive real numbers such that
    Sigma: x <= 1
    ^any finite subset of A

    Show A is finite or countably infinite

    Any help is much appreciated

    Consider the sets: S_n=\{a\in A: a>1/n\}\subseteq A where n \in \mathbb{N}

    S_n is clearly finite, and A=\bigcup_{n\in \mathbb{N}}S_n.

    So what do we know about countable unions of finite sets?

    CB
    Last edited by CaptainBlack; September 14th 2011 at 06:20 AM.
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