The Stacks project

Lemma 30.26.6. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $Z_ i \subset X$, $i = 1, \ldots , n$ be closed subsets. If $Z_ i$, $i = 1, \ldots , n$ are proper over $S$, then the same is true for $Z_1 \cup \ldots \cup Z_ n$.

Proof. Endow $Z_ i$ with their reduced induced closed subscheme structures. The morphism

\[ Z_1 \amalg \ldots \amalg Z_ n \longrightarrow X \]

is finite by Morphisms, Lemmas 29.44.12 and 29.44.13. As finite morphisms are universally closed (Morphisms, Lemma 29.44.11) and since $Z_1 \amalg \ldots \amalg Z_ n$ is proper over $S$ we conclude by Lemma 30.26.5 part (2) that the image $Z_1 \cup \ldots \cup Z_ n$ is proper over $S$. $\square$

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