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Math Help - Bolzano Weierstrass proof

  1. #1
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    Bolzano Weierstrass proof

    I have been asked to prove that a metric space has the Bolzano Weierstrass property if and only if it is complete and totally bounded.

    It seems obvious to me that Bolzano Weierstrass implies completeness however unless I am mistaken the real line is a counterexample (with the Euclidean metric) to the statement.

    R is not totally bounded and it satisfies the Bolzano Weierstrass property. Am I being stupid here? It is just that I was trying to do a past-paper question and this is what I am being asked to show.
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  2. #2
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    Quote Originally Posted by ihateyouall View Post
    I have been asked to prove that a metric space has the Bolzano Weierstrass property if and only if it is complete and totally bounded.

    It seems obvious to me that Bolzano Weierstrass implies completeness however unless I am mistaken the real line is a counterexample (with the Euclidean metric) to the statement.

    R is not totally bounded and it satisfies the Bolzano Weierstrass property. Am I being stupid here? It is just that I was trying to do a past-paper question and this is what I am being asked to show.
    The Bolzano–Weierstrass property, otherwise known as sequential compactness, is the property that every sequence has a convergent subsequence. The real line does not have this property. For example, the sequence of natural numbers does not have a convergent subsequence.
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  3. #3
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    Thanks. I knew that I must be wrong about something.
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