Lemma 25.23.8. An immersion of schemes is a monomorphism. In particular, any immersion is separated.

**Proof.**
We can see this by checking that the criterion of Lemma 25.23.7 applies. More elegantly perhaps, we can use that Lemmas 25.3.5 and 25.4.6 imply that open and closed immersions are monomorphisms and hence any immersion (which is a composition of such) is a monomorphism.
$\square$

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