I have the following question after reading Chapter V, prop. 4.3 of Miranda's book Algebraic Curves and Riemann Surfaces.
The setting is as follows: we have a Riemann surface $X$ and a holomorphic map $\phi: X\to \mathbb P^n$. We let $(x_0:\cdots : x_n)$ be the homogeneous coordinates on $\mathbb P^n$. The goal is to prove that $\phi$ can be defined by an $(n+1)$-tuple of meromorphic functions on $X$.
He proceeds as follows: assume that $x_0$ is not identically zero on $\phi(X)$ and define $f_i$ on $X$ to be the composition of $\phi$ with the function $x_i/x_0$. Next, he wants to prove that every such $f_i$ is meromorphic on $X$: locally around a point $p\in X$ the map $\phi$ is given by $\phi(z) = (g_0(z):\cdots :g_n(z))$ where, after fixing some local coordinate $z$ at $p$, each $g_i$ is a holomorphic function of $z$.
Then he says: note that $g_0$ is not identically zero near $p$. This should apparently contradict the fact that $x_0$ is not identically zero on $\phi(X)$.
What is the easiest way to make this completely rigorous? Thanks in advance!