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?