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Math Help - Is a compact subset A of a discrete metric space a finite set?

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    Is a compact subset A of a discrete metric space a finite set?

    Question: Prove that a compact subset A of a discrete metric space is a finite set.

    My Attempt: A is a compact subset of a discrete metric space (S,d). Therefore A is both closed and bounded. Since A is bounded there is some Neighborhood N_r(p) such that A is contained in the neighborhood, A \subseteq N_r(p). so, A \subseteq \{q \in S:d(p,q)<r \} ...

    And this is where I am getting lost. I know that I have to use the discrete metric d(p,q) = \left \{ \begin{array}{cc}0, & \mbox{ if } p=q\\1, & \mbox{ if } p!=q\end{array}\right. but I'm not sure how because I have never used a discrete metric before.

    Any hints, suggestions, or pushes in the right direction would be greatly appreciated.
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    Re: Is a compact subset A of a discrete metric space a finite set?

    Quote Originally Posted by SleepyGoose View Post
    Question: Prove that a compact subset A of a discrete metric space is a finite set.
    My Attempt: A is a compact subset of a discrete metric space (S,d). Therefore A is both closed and bounded. Since A is bounded there is some Neighborhood N_r(p) such that A is contained in the neighborhood, A \subseteq N_r(p). so, A \subseteq \{q \in S:d(p,q)<r \} ...
    For each a\in A define O_a=\{x:d(x,a)<0.5\}.

    For each O_a is open set that contains only one point.
    But \{O_a:a\in A\} is an open covering of A.

    Can you finish?
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