I am reading the text on Riemann surfaces by Farkas and Kra, and I'm having trouble understanding the following step in the Proposition on p. 12.
...it follows (even if the points $P_j$ are not distinct) that $\sum \limits_{P \in f^{-1}(Q)} (b_f(P) + 1) \geq n$.
Clearly, this is true if the $P_j$ are all distinct. However, I do not see the case when a group of say, $k$ of them are not distinct. I would need to show that the ramification number of $Q$ is $m$. In coordinates, this reduces to the following, which I don't see how to show.
Let $f: \mathbb{C} \to \mathbb{C}$ be analytic in a neighborhood of zero. Let $\{z_n\}$ be a sequence of complex numbers converging to zero. For each $n$, let $p_n^1, \ldots, p_n^k$ be distinct complex numbers such that $f(p_n^j) = z_n$, and $\lim \limits_{n \to \infty} z_n^j = 0$ for all $1 \leq j \leq k$. Then $f$ has a zero of mutiplicity $k$ at $z= 0$.