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    Real analysis - Proving Closure of a Ball

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    Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.

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    Solution Preview

    There is a choice of equivalent definitions of a closed set to use for the proof
    (see for instance http://mathworld.wolfram.com/ClosedSet.html).

    I suggest to use the following definition (4th in the page recommended) as the most convenient for the proof requested:

    A set S is closed if every point outside S has a neighborhood disjoint from (outside of) S.

    Now we look at the closed ball. Any point y outside of B[x;r] has

    d(x,y) > r (1)

    (by the definition of ...

    Solution Summary

    The solution proves closure of a ball.