Proof. This is implied by Algebra, Lemma 10.121.4 and Lemma 29.19.9. Alternatively, all points in fibres are closed points by Lemma 29.42.8 (and the fact that a finite morphism is integral) and use Lemma 29.19.6 (3) to see that $f$ is quasi-finite at $x$ for all $x \in X$. $\square$

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