I'm reading O. Debarre's Higher-dimensional algebraic geometry, and in Lemma 1.15 there is an argument, that a proper surjective morphism $p: Z \to Y$ of varieties, which has a finite fiber over a point $y_0 \in Y$ is actually a finite morphism over a small neighbourhood $Y_0 \subset Y$ of $y_0$.
I wonder why this is true. I think it suffices to show that $p$ has finite fibers in a neighbourhood of $y_0$, because properness is local in the base, and proper + quasi-finite implies finite (Hartshorne, Exercise III 11.2).
But why is it true that $p$ has finite fibers in a neighbourhood of $y_0$?