In a metric space $X$, or just $R^K$, $E \subset X$ is a closed set if every limit point of $E$ is a point of $E$. Is it possible to use only this fact, or definition, to prove that a finite point set is closed?
I know we can prove that because the complement of a finite point set is open. However, since a finite point set has no limit points as every neighborhood of a limit point should contain infinitely many points of $E$, I cannot figure out how to use this concept of limit points to prove that a finite point set is closed.
Thank you in advance!