A finite set is closed Question:

Prove that a finite subset in a metric space is closed.

My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next?
Definition: Set $E$ is called closed set if $E$ contains all his limits points.

Context: Principles of Mathematical Analysis, Rudin

If $M$ is a metric space then every subset $A =\{x_1, \ldots, x_n\} \subseteq M$ is closed. In fact, if $a \notin A$ then $d(a,A)$ is the least of the numbers $d(a,x_1) ,\ldots, d(a,x_n)$ thus, $d(a,A) > 0$. 
