This is prop 2.6b on p.28 of Silverman's the Arithmetic of Elliptic curves.

It says that let $\phi: C_1 \rightarrow C_2$ be a non-constant map of projective non-singular irreducible curve. (probably over an algebraically closed field, but I am not too sure) Then for all but finitely many $Q \in C_2$, #$\phi^{-1} (Q) = deg_s (\phi)$, where RHS is the separability degree of the function fields.

I don't understand Silverman's proof.

The proof just says that it is Hartshorne II.6.8, and I don't understand how it is related to this proposition at all. Hartshorne II.6.8 roughly states that if $f: X \rightarrow Y$ is a morphism where $X$ is a complete nonsingular curve over an algebraically closed field $k$, and $Y$ is any curve over $k$, then either $f(X) = pt$ or $Y$, and in the latter case, $f$ is finite morphism and $[K(X):K(Y)] < \infty$.

Can anyone show a proof of the proposition?

I failed to show that the set of all such $Q$ is open myself, can anyone shed some light on this? Thanks!