Originally Posted by

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

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