Let (X,d) have the property that every open cover of X has a finite subcover.

I want to show X is compact.

My start so far:

If X is not compact there exist a sequence $\displaystyle (x^{(n)})^{\infty}_{n=1}$ with no limit points. Then for every $\displaystyle x \in X$ there exists a ball $\displaystyle B(x,\epsilon)$ containing x which contains at most finitely many elements of this sequence..