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

My Attempt: is a compact subset of a discrete metric space . Therefore is both closed and bounded. Since is bounded there is some Neighborhood such that is contained in the neighborhood, . so, ...

And this is where I am getting lost. I know that I have to use the discrete metric 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.