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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 Summary

The solution proves closure of a ball.

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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:

DEFINITION:
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 ...

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