The Stacks project

Proof. Using Limits, Proposition 32.11.2 there is an immediate reduction to the case where $X$ is reduced. Details omitted. In the rest of the proof we endow every closed subset of $X$ with the induced reduced closed subscheme structure.

We argue by induction that we can find an affine open $U \subset Z_1 \cup \ldots \cup Z_ r$ containing $T \cap (Z_1 \cup \ldots \cup Z_ r)$. For $r = 1$ this holds by assumption. Say $r > 1$ and let $U \subset Z_1 \cup \ldots \cup Z_{r - 1}$ be an affine open containing $T \cap (Z_1 \cup \ldots \cup Z_{r - 1})$. Let $V \subset X_ r$ be an affine open containing $T \cap Z_ r$ (exists by assumption). Then $U \cap V$ contains $T \cap ( Z_1 \cup \ldots \cup Z_{r - 1} ) \cap Z_ r$. Hence

does not contain any element of $T$. Note that $\Delta$ is a closed subset of $U$. By prime avoidance (Algebra, Lemma 10.15.2), we can find a standard open $U'$ of $U$ containing $T \cap U$ and avoiding $\Delta$, i.e., $U' \cap Z_ r \subset U \cap V$. After replacing $U$ by $U'$ we may assume that $U \cap V$ is closed in $U$.

Using that by the same arguments as above also the set $\Delta ' = (U \cap (Z_1 \cup \ldots \cup Z_{r - 1})) \setminus (U \cap V)$ does not contain any element of $T$ we find a $h \in \mathcal{O}(V)$ such that $D(h) \subset V$ contains $T \cap V$ and such that $U \cap D(h) \subset U \cap V$. Using that $U \cap V$ is closed in $U$ we can use Lemma 33.42.5 to find an element $g \in \mathcal{O}(U)$ whose restriction to $U \cap V$ equals the restriction of $h$ to $U \cap V$ and such that $T \cap U \subset D(g)$. Then we can replace $U$ by $D(g)$ and $V$ by $D(h)$ to reach the situation where $U \cap V$ is closed in both $U$ and $V$. In this case the scheme $U \cup V$ is affine by Limits, Lemma 32.11.3. This proves the induction step and thereby the lemma. $\square$