Lemma 29.43.11. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$f$ is finite, and

$f$ is affine and proper.

Lemma 29.43.11. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

$f$ is finite, and

$f$ is affine and proper.

**Proof.**
This follows formally from Lemma 29.43.7, the fact that a finite morphism is integral and separated, the fact that a proper morphism is the same thing as a finite type, separated, universally closed morphism, and the fact that an integral morphism of finite type is finite (Lemma 29.43.4).
$\square$

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