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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