Choose $d_0$ for $X_1 \to S_1$ and $\mathcal{L}_1$ as in More on Morphisms, Lemma \ref{more-morphisms-lemma-deform-projective}. For any $n \geq 1$ set $$ A_n = R_n \oplus \bigoplus\nolimits_{d \geq d_0} H^0(X_n, \mathcal{L}_n^{\otimes d}) $$ By the lemma each $A_n$ is a finitely presented graded $R_n$-algebra whose homogeneous parts $(A_n)_d$ are finite projective $R_n$-modules such that $X_n = \text{Proj}(A_n)$ and $\mathcal{L}_n = \mathcal{O}_{\text{Proj}(A_n)}(1)$. The lemma also guarantees that the maps $$ A_1 \leftarrow A_2 \leftarrow A_3 \leftarrow \ldots $$ induce isomorphisms $A_n = A_m \otimes_{R_m} R_n$ for $n \leq m$. We set $$ B = \bigoplus\nolimits_{d \geq 0} B_d \quad\text{with}\quad B_d = \lim_n (A_n)_d $$ By More on Algebra, Lemma \ref{more-algebra-lemma-lim-finite-projective-gives-finite-projective} we see that $B_d$ is a finite projective $R$-module and that $B \otimes_R R_n = A_n$. Thus the scheme $$ X = \text{Proj}(B) \quad\text{and}\quad \mathcal{L} = \mathcal{O}_X(1) $$ is flat over $S$ and $\mathcal{L}$ is a quasi-coherent $\mathcal{O}_X$-module flat over $S$, see Divisors, Lemma \ref{divisors-lemma-relative-proj-flat}. Because formation of Proj commutes with base change (Constructions, Lemma \ref{constructions-lemma-base-change-map-proj}) we obtain canonical isomorphisms $$ X \times_S S_n = X_n \quad\text{and}\quad \mathcal{L}|_{X_n} \cong \mathcal{L}_n $$ compatible with the transition maps of the system. Thus we may think of $X_1 \subset X$ as a closed subscheme. Below we will show that $B$ is of finite presentation over $R$. By Divisors, Lemmas \ref{divisors-lemma-relative-proj-proper} and \ref{divisors-lemma-relative-proj-finite-presentation} this implies that $X \to S$ is of finite presentation and proper and that $\mathcal{L} = \mathcal{O}_X(1)$ is of finite presentation as an $\mathcal{O}_X$-module. Since the restriction of $\mathcal{L}$ to the base change $X_1 \to S_1$ is invertible, we see from More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-free-open} that $\mathcal{L}$ is invertible on an open neighbourhood of $X_1$ in $X$. Since $X \to S$ is closed and since $\Ker(R \to R_1)$ is contained in the Jacobson radical (More on Algebra, Lemma \ref{more-algebra-lemma-limit-henselian}) we see that any open neighbourhood of $X_1$ in $X$ is equal to $X$. Thus $\mathcal{L}$ is invertible. Finally, the set of points in $S$ where $\mathcal{L}$ is ample on the fibre is open in $S$ (More on Morphisms, Lemma \ref{more-morphisms-lemma-ample-in-neighbourhood}) and contains $S_1$ hence equals $S$. Thus $X \to S$ and $\mathcal{L}$ have all the properties required of them in the statement of the lemma. \medskip\noindent We prove the claim above. Choose a presentation $A_1 = R_1[X_1, \ldots, X_s]/(F_1, \ldots, F_t)$ where $X_i$ are variables having degrees $d_i$ and $F_j$ are homogeneous polynomials in $X_i$ of degree $e_j$. Then we can choose a map $$ \Psi : R[X_1, \ldots, X_s] \longrightarrow B $$ lifting the map $R_1[X_1, \ldots, X_s] \to A_1$. Since each $B_d$ is finite projective over $R$ we conclude from  Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK} using again that $\Ker(R \to R_1)$ is contained in the Jacobson radical of $R$) that $\Psi$ is surjective. Since $- \otimes_R R_1$ is right exact we can find $G_1, \ldots, G_t \in \Ker(\Psi)$ mapping to $F_1, \ldots, F_t$ in $R_1[X_1, \ldots, X_s]$. Observe that $\Ker(\Psi)_d$ is a finite projective $R$-module for all $d \geq 0$ as the kernel of the surjection $R[X_1, \ldots, X_s]_d \to B_d$ of finite projective $R$-modules. We conclude from Nakayama's lemma once more that  $\Ker(\Psi)$ is generated by $G_1, \ldots, G_t$.