Since the squares in the diagram are cartesian and since the morphisms $S_n \to S_{n + 1}$ are closed immersions, we see that the morphisms $i_n$ are closed immersions too. In particular we may think of $X_m$ as a closed subscheme of $X_n$ for $m < n$. In fact $X_m$ is the closed subscheme cut out by the quasi-coherent sheaf of ideals $I^m\mathcal{O}_{X_n}$. Moreover, the underlying topological spaces of the schemes $X_1, X_2, X_3, \ldots$ are all identified, hence we may (and do) think of sheaves $\mathcal{O}_{X_n}$ as living on the same underlying topological space; similarly for coherent $\mathcal{O}_{X_n}$-modules. Set $$ \mathcal{F}_n = \Ker(\mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_n}) $$ so that we obtain short exact sequences $$ 0 \to \mathcal{F}_n \to \mathcal{O}_{X_{n + 1}} \to \mathcal{O}_{X_n} \to 0 $$ By the above we have $\mathcal{F}_n = I^n\mathcal{O}_{X_{n + 1}}$. It follows $\mathcal{F}_n$ is a coherent sheaf on $X_{n + 1}$ annihilated by $I$, hence we may (and do) think of it as a coherent module $\mathcal{O}_{X_1}$-module. Observe that for $m > n$ the sheaf $$ I^n\mathcal{O}_{X_m}/I^{n + 1}\mathcal{O}_{X_m} $$ maps isomorphically to $\mathcal{F}_n$ under the map $\mathcal{O}_{X_m} \to \mathcal{O}_{X_{n + 1}}$. Hence given $n_1, n_2 \geq 0$ we can pick an $m > n_1 + n_2$ and consider the multiplication map $$ I^{n_1}\mathcal{O}_{X_m} \times I^{n_2}\mathcal{O}_{X_m} \longrightarrow I^{n_1 + n_2}\mathcal{O}_{X_m} \to \mathcal{F}_{n_1 + n_2} $$ This induces an $\mathcal{O}_{X_1}$-bilinear map $$ \mathcal{F}_{n_1} \times \mathcal{F}_{n_2} \longrightarrow \mathcal{F}_{n_1 + n_2} $$ which in turn defines the structure of a graded $\mathcal{O}_{X_1}$-algebra on $\mathcal{F} = \bigoplus_{n \geq 0} \mathcal{F}_n$. \medskip\noindent Set $B = \bigoplus I^n/I^{n + 1}$; this is a finitely generated graded $A/I$-algebra. Set $\mathcal{B} = (X_1 \to S_1)^*\widetilde{B}$. The discussion above provides us with a canonical surjection $$ \mathcal{B} \longrightarrow \mathcal{F} $$ of graded $\mathcal{O}_{X_1}$-algebras. In particular we see that $\mathcal{F}$ is a finite type quasi-coherent graded $\mathcal{B}$-module. By Lemma \ref{lemma-graded-finiteness} we can find an integer $d_0$ such that $H^1(X_1, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0$ for all $d \geq d_0$. Pick a $d \geq d_0$ such that there exist sections $s_{0, 1}, \ldots, s_{N, 1} \in \Gamma(X_1, \mathcal{L}_1^{\otimes d})$ which induce an immersion $$ \psi_1 : X_1 \to \mathbf{P}^N_{S_1} $$ over $S_1$, see Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}. As $X_1$ is proper over $S_1$ we see that $\psi_1$ is a closed immersion, see Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed} and Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}. We are going to ``lift'' $\psi_1$ to a compatible system of closed immersions of $X_n$ into $\mathbf{P}^N$. \medskip\noindent Upon tensoring the short exact sequences of the first paragraph of the proof by $\mathcal{L}_{n + 1}^{\otimes d}$ we obtain short exact sequences $$ 0 \to \mathcal{F}_n \otimes \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to 0 $$ Using the isomorphisms $\varphi_n$ we obtain isomorphisms $\mathcal{L}_{n + 1} \otimes \mathcal{O}_{X_l} = \mathcal{L}_l$ for $l \leq n$. Whence the sequence above becomes $$ 0 \to \mathcal{F}_n \otimes \mathcal{L}_1^{\otimes d} \to \mathcal{L}_{n + 1}^{\otimes d} \to \mathcal{L}_n^{\otimes d} \to 0 $$ The vanishing of $H^1(X, \mathcal{F}_n \otimes \mathcal{L}_1^{\otimes d})$ implies we can inductively lift $s_{0, 1}, \ldots, s_{N, 1} \in \Gamma(X_1, \mathcal{L}_1^{\otimes d})$ to sections $s_{0, n}, \ldots, s_{N, n} \in \Gamma(X_n, \mathcal{L}_n^{\otimes d})$. Thus we obtain a commutative diagram $$ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d]_{\psi_1} & X_2 \ar[r]_{i_2} \ar[d]_{\psi_2} & X_3 \ar[r] \ar[d]_{\psi_3} & \ldots \\ \mathbf{P}^N_{S_1} \ar[r] & \mathbf{P}^N_{S_2} \ar[r] & \mathbf{P}^N_{S_3} \ar[r] & \ldots } $$ where $\psi_n = \varphi_{(\mathcal{L}_n, (s_{0, n}, \ldots, s_{N, n}))}$ in the notation of Constructions, Section \ref{constructions-section-projective-space}. As the squares in the statement of the theorem are cartesian we see that the squares in the above diagram are cartesian. We win by applying Lemma \ref{lemma-algebraize-formal-closed-subscheme}.