# Proving that a finite point set is closed by using limit points

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.

• If $E$ has no limit points, then all of them are vacuously elements of $E.$ – saulspatz Oct 20 '18 at 17:43

Proposition: If a set $$X$$ in a metric space has a limit point $$x$$, then $$X$$ is infinite.
Proof. Consider the ball of radius $$r=1$$ centred at $$x$$. By our assumption, this ball contains a point in $$X$$ that is distinct from $$x$$. Call this point $$x_{1}$$. Now, consider the ball of radius $$r=d(x,x_{1})$$ centred at $$x$$. Again, this ball contains a point in $$X$$ that is distinct from $$x$$. Moreover, this new point cannot equal to $$x_{1}$$ due to our choice of radius. Call this new point $$x_{2}$$. Continuing this way, we obtain a sequence of points in $$X$$ which are all distinct from each other. Therefore, $$X$$ is infinite.