I am trying to do an exercise from Rick Miranda's Book that goes like this
Let $X$ be an algebraic Curve, show using the compactness of $X$ that there are a finite number of global meromorphic functions on $X$ wich separate the points and tangents of $X$.
So I think I was able to do this for the tangents because the condition that it separates tangents is equivalent that any point $p$ there is a local coordinate wich extends to a meromorphic function on all of $X$, so we can take the neighborhoods associated to it being a coordinate chart and cover $X$ and then we get a finite number of coordinate charts and hence a finite number of functions that separate the tangents.
Now for the point I still haven't managed to do it I tried doing something similar as the previous one but, cause we know that for every point the pre-image of $f(p)$ will be finite since this $X$ is compact but here I am initially fixing $p$ so I still haven't manage to figure out how to get something finite. I just can't seem to get anywhere. Any help is aprecciated.
Thanks in advance.