Lemma 29.2.5. A composition of closed immersions is a closed immersion.

**Proof.**
We have seen this in Schemes, Lemma 26.24.3, but here is another proof. Namely, it follows from the characterization (3) of closed immersions in Lemma 29.2.1. Since if $I \subset R$ is an ideal, and $\overline{J} \subset R/I$ is an ideal, then $\overline{J} = J/I$ for some ideal $J \subset R$ which contains $I$ and $(R/I)/\overline{J} = R/J$.
$\square$

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