Lemma 29.10.3. A universally injective morphism is separated.

**Proof.**
Combine Lemma 29.10.2 with the remark that $X \to S$ is separated if and only if the image of $\Delta _{X/S}$ is closed in $X \times _ S X$, see Schemes, Definition 26.21.3 and the discussion following it.
$\square$

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