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Math Help - prove M is complete

  1. #1
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    prove M is complete

    Suppose that every countable, closed subset of M is complete. Prove that M is complete.


    my idea is to show that M is closed first, then let (Xn) be a cauchy sequence in M, so it's cauchy in all of its subset and then converge, so M is complete.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: prove M is complete

    Quote Originally Posted by wopashui View Post
    Suppose that every countable, closed subset of M is complete. Prove that M is complete.


    my idea is to show that M is closed first, then let (Xn) be a cauchy sequence in M, so it's cauchy in all of its subset and then converge, so M is complete.
    But, M may not be countable, so this doesn't work. Try proving that if (x_n) is a Cauchy sequence then X=\overline{\{x_n:n\in\mathbb{N}\}} contains at most one more element and thus is also countable. But, then (x_n) is a Cauchy sequence in the countable closed subset X\subseteq M.
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  3. #3
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    Re: prove M is complete

    if A is subset of M, and (Xn) is cauchy in M, does it imply (Xn) is cauchy in A?
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    MHF Contributor Drexel28's Avatar
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    Re: prove M is complete

    Quote Originally Posted by wopashui View Post
    if A is subset of M, and (Xn) is cauchy in M, does it imply (Xn) is cauchy in A?
    Yes. But, the problem is that you need to prove you can always put (x_n) inside some COUNTABLE closed subset.
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