Let $A = \Gamma_*(X, \mathcal{L}) = \bigoplus_{d \geq 0} \Gamma(X, \mathcal{L}^{\otimes d})$. By Properties, Proposition \ref{properties-proposition-characterize-ample} the set of affine opens $X_a$ with $a \in A_{+}$ homogeneous forms a basis for the topology of $X$. Hence we can find finitely many such elements $a_0, \ldots, a_n \in A_{+}$ such that \begin{enumerate} \item we have $X = \bigcup_{i = 0, \ldots, n} X_{a_i}$, \item each $X_{a_i}$ is affine, and \item each $X_{a_i}$ maps into an affine open $V_i \subset S$. \end{enumerate} By Lemma \ref{lemma-locally-finite-type-characterize} we see that the ring maps $\mathcal{O}_S(V_i) \to \mathcal{O}_X(X_{a_i})$ are of finite type. Hence we can find finitely many elements $f_{ij} \in \mathcal{O}_X(X_{a_i})$, $j = 1, \ldots, n_i$ which generate $\mathcal{O}_X(X_{a_i})$ as an $\mathcal{O}_S(V_i)$-algebra. By Properties, Lemma \ref{properties-lemma-invert-s-sections} we may write each $f_{ij}$ as $a_{ij}/a_i^{e_{ij}}$ for some $a_{ij} \in A_{+}$ homogeneous. Let $N$ be a positive integer which is a common multiple of all the degrees of the elements $a_i$, $a_{ij}$. Consider the elements $$ a_i^{N/\deg(a_i)}, \ a_{ij}a_i^{(N/\deg(a_i)) - e_{ij}} \in A_N. $$ By construction these generate the invertible sheaf $\mathcal{L}^{\otimes N}$ over $X$. Hence they give rise to a morphism $$ j : X \longrightarrow \mathbf{P}_S^{m} \quad \text{with } m = n + \sum n_i $$ over $S$, see Constructions, Lemma \ref{constructions-lemma-projective-space} and Definition \ref{constructions-definition-projective-space}. Moreover, $j^*\mathcal{O}_{\mathbf{P}_S}(1) = \mathcal{L}^{\otimes N}$. We name the homogeneous coordinates $T_0, \ldots, T_n, T_{ij}$ instead of $T_0, \ldots, T_m$. For $i = 0, \ldots, n$ we have $i^{-1}(D_{+}(T_i)) = X_{a_i}$. Moreover, pulling back the element $T_{ij}/T_i$ via $j^\sharp$ we get the element $f_{ij} \in \mathcal{O}_X(X_{a_i})$. Hence the morphism $j$ restricted to $X_{a_i}$ gives a closed immersion of $X_{a_i}$ into the affine open $D_{+}(T_i) \cap \mathbf{P}^m_{V_i}$ of $\mathbf{P}^N_S$. Hence we conclude that the morphism $j$ is an immersion. This implies the lemma holds for some $d$ and $n$ which is enough in virtually all applications. \medskip\noindent This proves that for one $d_2 \geq 1$ (namely $d_2 = N$ above), some $m \geq 0$ there exists some immersion $j : X \to \mathbf{P}^m_S$ given by global sections $s'_0, \ldots, s'_m \in \Gamma(X, \mathcal{L}^{\otimes d_2})$. By Properties, Proposition \ref{properties-proposition-characterize-ample} we know there exists an integer $d_1$ such that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq d_1$. Set $d_0 = d_1 + d_2$. We claim that the lemma holds with this value of $d_0$. Namely, given an integer $d \geq d_0$ we may choose $s''_1, \ldots, s''_t \in \Gamma(X, \mathcal{L}^{\otimes d - d_2})$ which generate $\mathcal{L}^{\otimes d - d_2}$ over $X$. Set $k = (m + 1)t$ and denote $s_0, \ldots, s_k$ the collection of sections $s'_\alpha s''_\beta$, $\alpha = 0, \ldots, m$, $\beta = 1, \ldots, t$. These generate $\mathcal{L}^{\otimes d}$ over $X$ and therefore define a morphism $$ i : X \longrightarrow \mathbf{P}^{k - 1}_S $$ such that $i^*\mathcal{O}_{\mathbf{P}^n_S}(1) \cong \mathcal{L}^{\otimes d}$. To see that $i$ is an immersion, observe that $i$ is the composition $$ X \longrightarrow \mathbf{P}^m_S \times_S \mathbf{P}^{t - 1}_S \longrightarrow \mathbf{P}^{k - 1}_S $$ where the first morphism is $(j, j')$ with $j'$ given by $s''_1, \ldots, s''_t$ and the second morphism is the Segre embedding (Constructions, Lemma \ref{constructions-lemma-segre-embedding}). Since $j$ is an immersion, so is $(j, j')$ (apply Lemma \ref{lemma-immersion-permanence} to $X \to \mathbf{P}^m_S \times_S \mathbf{P}^{t - 1}_S \to \mathbf{P}^m_S$). Thus $i$ is a composition of immersions and hence an immersion (Schemes, Lemma \ref{schemes-lemma-composition-immersion}).