The Stacks project

Proof. By the discussion in Remark 32.22.5 the choice of $i_1$ and $W$ fitting into a diagram as in (32.22.2.1) is immaterial for the truth of the lemma. Thus we choose $W$ as follows. First we choose a closed immersion $X \to X'$ with $X' \to S$ proper and of finite presentation, see Lemma 32.13.2. Then we choose an $i_3 \geq i_2$ and a proper morphism $W \to Y_{i_3}$ such that $X' = Y \times _{Y_{i_3}} W$. This is possible because $Y = \mathop{\mathrm{lim}}\nolimits _{i \geq i_2} Y_ i$ and Lemmas 32.10.1 and 32.13.1. With this choice of $W$ it is immediate from the construction that for $i \geq i_3$ the scheme $X_ i$ is a closed subscheme of $Y_ i \times _{Y_{i_3}} W \subset S_ i \times _{S_{i_3}} W$ and hence proper over $Y_ i$. $\square$