An Elementary Approach to How Many Singular Points a Curve Can Have Consider an irreducible plane curve, $C \subseteq A^{2}_{\mathbb{C}}$. Prove that C has only finitely many singular points. 
My idea is to consider the subset $S$ that contains the singular points of C. Then we would want to show that it is a closed subset--but I do not see where to go from here. Could someone help, with elementary methods? 

Hint: The singular points should satisfy some equation, and so should be Zariski closed. That means they are either everything or finite (why?). But, they can't be everything (why?).
