Finite Morphisms of Affine Curves are Surjective

Question

I have a question about a step used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 109):

The statement is that a finite morphism of integral affine curves is surjective.

Indeed I'm familar with some proofs to show this statement but the point I'm really interested is concretely the argument given in this source by the author.

He claims that the Case 2 of the proof of 4.3.5 cannot occure in the case of a finite morphism.

My question is why?

Here the proof of 4.3.5:

Answer 1

The argument in the book is too complicated:
A finite morphism of schemes is a closed map, hence the image of Y is closed and irreducible in $X$ and thus that image is equal to $X$.
By the way, my argument does not use that $X$ or $Y$ is affine, nor that $Y$ has dimension $1$!