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Math Help - Metric Spaces

  1. #1
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    Metric Spaces

    Need some help.

    1. Prove that every finite subset of a metric space, X, is closed.
    On this, am I trying to prove that X is complete by showing that every Cauchy sequence converges to some point in X?


    2. How do I prove that Q is not closed in R?

    Thanks.
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  2. #2
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    Quote Originally Posted by taypez View Post
    1. Prove that every finite subset of a metric space, X, is closed.
    2. How do I prove that Q is not closed in R?
    “On this, am I trying to prove that X is complete by showing that every Cauchy sequence converges to some point in X?” has nothing to do with #1.
    If F is a finite subset of a metric space X just show that the complement of F is an open set. For any point t not in F it is easy to construct an ball centered at t containing no other point of F.

    For #2, you know that every irrational number is the limit of a sequence of rational numbers. Put another way: Every irrational number is a limit point of the set of rational numbers.
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