Proof. Let $f : X \to S$ be locally projective. In order to show that $f$ is proper we may work locally on the base, see Lemma 29.40.3. Hence, by Lemma 29.42.4 above we may assume there exists a closed immersion $X \to \mathbf{P}^ n_ S$. By Lemmas 29.40.4 and 29.40.6 it suffices to prove that $\mathbf{P}^ n_ S \to S$ is proper. Since $\mathbf{P}^ n_ S \to S$ is the base change of $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ it suffices to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is proper, see Lemma 29.40.5. By Constructions, Lemma 27.8.8 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is separated. By Constructions, Lemma 27.8.9 the scheme $\mathbf{P}^ n_{\mathbf{Z}}$ is quasi-compact. It is clear that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is locally of finite type since $\mathbf{P}^ n_{\mathbf{Z}}$ is covered by the affine opens $D_{+}(X_ i)$ each of which is the spectrum of the finite type $\mathbf{Z}$-algebra

$\mathbf{Z}[X_0/X_ i, \ldots , X_ n/X_ i].$

Finally, we have to show that $\mathbf{P}^ n_{\mathbf{Z}} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is universally closed. This follows from Constructions, Lemma 27.8.11 and the valuative criterion (see Schemes, Proposition 26.20.6). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).