Finite Number of Global Meromorphic Functions Separating Points and Tangents on a Compact Riemann Surface In Miranda's "Algebraic Curves and Riemann Surfaces," there is a problem that asks the reader the show, using the compactness of a Riemann Surface $X$, that there are only a finite number of global meromorphic functions separating points and tangents on $X$? We are assuming that the field of rational functions on $X$ separates points and tangents, so there's no need to show any existence.
I've been interpreting this question to mean that there are only a finite number of global meromorphic functions on $X$ which take different values at every point on $X$ and for which, at every point $p$ in $X$, the function is either holomorphic at $p$ with the order of $f - f(p)$ at $p$ equal to one or has a simple pole at $p$, using the definitions from the text about what it means for the entire field of meromorphic functions to separate points and tangents.
Is this interpretation accurate, or am I misreading? Given that my interpretation is accurate (and I suppose even if it's not), what is the quickest solution? 
 A: I belive the idea is the following :
So let $X$ be the algebraic curve, first to see that it there are a finite number of functions that separate tangents we notice that the condition for separating tangents is equivalent to the condition that at every $p$ there is a local coordinate chart around $V_p$ so you can take these open sets and cover $X$ and retrieve a finite number of them since $X$ is compact and we will have the finite number of functions associated to these open sets will do what we desire.
Now to see that it separates points we know that for every $x,y\in X$ there is $f_{x,y}$ such that it separates $x$ and $y$, and so since the function is continuous there are neighborhoods $V_x$ and $V_y$ such that $f_{x,y}$ separates $V_x$ from $V_y$, now the idea is to try and cover $X\times X$ with $V_{x,y}=V_x\times V_y$ the only problem is that this open sets don't get the diagonal and for this we need the separating tangents condition again to find $f_x$ such that is bijective in a neighborhood of $x$ $V_x$ and we consider $V_x\times V_x$. Now we take a finite subcover since $X\times X$ is compact and the meromorphic functions associated to each $V_{x,y}$ will separate the points.
