In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function $ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane $R(s) > dim(X)$, where $X$ is a scheme of finite type over $\mathbb{Z}$, $|X|$ the set of closed points of $X$ and $N(x)$ the number of elements in the residue field $k(x)$.
He reduces the claim to the case where $X = Spec \, A[x_1, \ldots x_n]$ and $A$ is either $\mathbb{Z}$ or $\mathbb{F}_p$.
The decisive input is the following lemma:
a) If $X$ is the finite union of the schemes $X_i$, and the claim holds for all $X_i$, then it holds for $X$.
b) If $f: X \to Y$ is finite and the claim holds for $Y$, then it holds for $X$ as well.
I've been trying to prove b) but I seem to be missing something. Here's what I've tried so far: I was considering $\zeta(X,s) = \prod_{y \in |Y|} \zeta(X_y \, ,s)$, where $X_y$ is the fiber of $f$ at $y$. I now the fibers are finite but I don't know how to connect this with the fact that $\zeta(Y,s)$ converges. Is it true that the residue field $k(y)$ is a finite extension of $k(x)$ for all $x \in X_y$ (of degree $\deg f$)? I know this is the case for the function fields.
Any help is very appreciated!