I'm reading "Riemann Surfaces" by Farkas and Kra. Section I.1.6 contains the following proposition:
Let $f : M \to N$ be a non-constant holomorphic mapping between compact Riemann surfaces. There exists a positive integer $m$ such that every $Q \in N$ is assumed precisely $m$ times on $M$ by $f$, counting multiplicities. That is, for all $Q \in N$, $$\sum_{P \in f^{-1}(Q)} [b_f(P) + 1] = m.$$
The proof begins like this:
For each integer $n \geq 1$, let $$ \Sigma_n := \left\{Q \in N \; \middle| \; \sum_{P \in f^{-1}(Q)} [b_f(P) + 1] \geq n \right\}$$ The "normal form" of the mapping $f$ given by (1.6.1) shows that $\Sigma_n$ is open in $N$. [...]
Here (1.6.1) is the result that an analytic map can locally be written as $w = z^n$ in appropriate coordinates.
I guess there must be a theorem from complex analysis saying that the zero locus of a holomorphic function is discrete; if so, that (and compactness of $M$) establishes that $f^{-1}(Q)$ is finite, say $f^{-1}(Q) = \{P_1, \ldots, P_k\}$. Now I guess I need to establish that there's some neighborhood $D \ni Q$ such that $f^{-1}(D) = D_1 \sqcup \cdots \sqcup D_k$, where $P_i \in D_i$, and such that the $D_i$ are small enough that each $f|_{D_i}$ can be written as $z \mapsto z^{n_i}$ after a change of coordinates.
Is this what the authors mean here? If so, how do I establish these facts? Do I have to use the normal form more than once in the argument? My concern is that maybe every open neighborhood of $Q$, no matter how small, could have a very large preimage.