Let $f : S_1 \to S_2$ be a non-constant holomorphic function of Riemann surfaces. Why doesn't the set $$\{p \in S_1 \mid v_f(p) > 1\}$$ has an accumulation point?
This should be related to the fact that any non-constant holomorphic function can be represented at a point by $z \mapsto z^n$, but I cannot see how!
Hint: $v_f(p)>1$ iff locally the derivative of $f$ vanishes at $p$.