The Stacks project

Lemma 29.41.6. A closed immersion is proper, hence a fortiori universally closed.

Proof. The base change of a closed immersion is a closed immersion (Schemes, Lemma 26.18.2). Hence it is universally closed. A closed immersion is separated (Schemes, Lemma 26.23.8). A closed immersion is of finite type (Lemma 29.15.5). Hence a closed immersion is proper. $\square$

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