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Math Help - closure problem

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    closure problem

    Let (X,p) be a metric space and let S X be totally bounded. Argue that the closure of S is totally bounded.
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    Quote Originally Posted by jburks100 View Post
    Let (X,p) be a metric space and let S I X be totally bounded. Argue that the closure of S is totally bounded.
    Suppose that \overline{S} is not totally bounded.
    There is c>0 such that there is no c\text{-net} for \overline{S}.
    But there must a c\text{-net} for {S}.
    So some y\in\overline{S} such that y is not in \bigcup\limits_{k = 1}^n {B(\alpha _k ;c)} the c\text{-net} for S.

    Let d=\min\{d(y,\alpha_k)-c\},~k=1,2,\cdots n.
    You need to argue that d>0.
    But B(y;d) must contain a point in S.
    Now prove that is a contradiction.
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