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Math Help - Cauchy sequence

  1. #1
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    Cauchy sequence

    Don't need a proof, just need confirmation.
    Every cauchy sequence converges. This is true right?
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  2. #2
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    If you are working in the real number field then that is correct.
    It is correct in any complete metric space.
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  3. #3
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    Quote Originally Posted by Plato View Post
    If you are working in the real number field then that is correct.
    It is correct in any complete metric space.
    I have a question.

    Complete, as in "completeness", as in in the order property of lub/glb?
    Then, every complete metric space is isomorphic to the real number field because there can only one complete ordered field up to isomorphism.
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  4. #4
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    Actually you are correct to ‘call my hand’ on that.
    In most treatments a complete space is one in which every Cauchy sequence converges. Thus, in some way my answer is circular.
    In my own defense, some texts that is equivalent to every infinite bounded set has a limit point.
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