For convenient we discuss about projective curves. Hartshorne II 6.8 proved a non-trivial morphism from a nonsingular curve to another curve is finite. So I'm wondering if we can prove the consequence for any curves and non-trivial morphism. I tried to do this by following: For any point y on Y, I take an affine neighborhood $V=\operatorname{Spec} A$ of it, and set $U=f^{-1}(V)$. The fiber is $\operatorname{Spec}(\mathcal{O}(U)\otimes_A k(y))$. $\mathcal{O}(U)\otimes_A k(y)$ is a finite length k(y)-module hence Artinian, so the fiber is finite and discrete, therefore $f$ is quasi-finite.

I know there is a consequence quasi-finite+proper=finite, but I have no ideal how to prove it(does anyone have a ref?) and I want to deal with the problem (in projective case) using a rather elementary method, but I cant think up one.