High powers of an ample line bundle embed a proper scheme

$X$ is quasi-compact: $S$ is Noetherian, hence quasi-compact, and $f$ is of finite type, hence quasi-compact, so $X=f^{-1}(S)$ is quasi-compact by the definition of a quasi-compact morphism.

$X$ is locally Noetherian: choose a finite affine open cover $\mathrm{Spec}{A}_{\gamma }$ of the Noetherian scheme $S$, with every $A_{\gamma }$ Noetherian [F2]. Each inverse image $X_{\gamma }=f^{-1}(\mathrm{Spec}{A}_{\gamma })$ is of finite type over $\mathrm{Spec}{A}_{\gamma }$ and hence quasi-compact [F2], so choose a finite affine open cover $X_{\gamma }={\bigcup }_i\mathrm{Spec}{B}_{\gamma i}$. For each chart the ring map $A_{\gamma }\to B_{\gamma i}$ is of finite type by the affine-local criterion [F2], and $B_{\gamma i}$ is Noetherian by [F3]. These finitely many Noetherian affine opens cover $X$, so $X$ is locally Noetherian.

Derived closure facts. (a) A composite of closed immersions is a closed immersion: closed immersions are local on the target, and over an affine target a closed immersion is up to unique isomorphism a quotient map $\mathrm{Spec}(A\mathrm{/}I)\to \mathrm{Spec}A$ [F9]; composing the quotients $A\to A\mathrm{/}I\to (A\mathrm{/}I)\mathrm{/}J\cong A\mathrm{/}{J}^{\mathrm{\prime }}$ exhibits the composite again as a quotient map, and locality on the target gives the general case. (b) If $\iota$ is an immersion and $c$ is a closed immersion with source the source of $\iota$, then $\iota \circ c$ is an immersion: writing $\iota =u\circ d$ with $d$ a closed immersion and $u$ an open immersion [F10], one has $\iota \circ c=u\circ (d\circ c)$ with $d\circ c$ a closed immersion by (a).

$X$ is quasi-separated: let $U,V\subseteq X$ be affine open. By step 1.2, $X$ has a finite open cover by spectra of Noetherian rings, each a Noetherian topological space by [F16]. A finite union of Noetherian open subspaces is Noetherian: an ascending chain of opens stabilizes on each member of the finite cover and therefore stabilizes on their union. Thus the underlying space of $X$ is Noetherian, and by [F17] its open subset $U\cap V$ is quasi-compact. As this holds for every pair of affine opens, [F15] makes $X$ quasi-separated.

$X$ is Noetherian: it is locally Noetherian by step 1.2 and quasi-compact by step 1.1.

The affine positive-degree nonvanishing loci form a basis. Let $U\subseteq X$ be open and $x\in U$. By [F1] there are $n_0\ge 1$ and $s_0\in \mathrm{\Gamma }(X,L^{\otimes n_0})$ with $x\in X_{s_0}$ and $X_{s_0}$ affine, say $X_{s_0}=\mathrm{Spec}C$. Since $U\cap X_{s_0}$ is an open neighbourhood of $x$ in that affine scheme, [F6] gives $h\in C$ with $x\in D(h)\subseteq U\cap X_{s_0}$, and $D(h)$ is affine. By step 1.1 $X$ is quasi-compact and by step 2.1 quasi-separated, so [F5] applied to the quasi-coherent sheaf ${\mathcal{O}}_X$ ([F4]), the invertible sheaf $L$, the integer $n_0>0$, the section $s_0$ and the section $h\in \mathrm{\Gamma }(X_{s_0},{\mathcal{O}}_X)$ yields $r\ge 0$ and $a\in \mathrm{\Gamma }(X,L^{\otimes n_{0}r})$ with $a\otimes s_0^{-r}=h$ on $X_{s_0}$; replacing $r$ by $r+1$ and $a$ by $a\otimes s_0$ if necessary we may assume $r\ge 1$. Then on $X_{s_0}$ the section $a$ restricts to $h{s}_0^r$, so $X_a\cap X_{s_0}=D(h)$, and the product $a\otimes s_0\in \mathrm{\Gamma }(X,L^{\otimes n_0(r+1)})$ satisfies $X_{a\otimes s_0}=X_a\cap X_{s_0}=D(h)$ [F1]. This locus contains $x$, lies in $U$, and is affine, so the sets $X_a$ with $a\in A_{+}$ form a basis for the topology of $X$.

Serre's bound. The structure sheaf ${\mathcal{O}}_X$ is invertible, hence quasi-coherent [F4], and it is of finite type because on every affine chart it is generated by its unit section; since $X$ is Noetherian by step 2.2 and $L$ is ample, the criterion [F4] provides an integer $d_1\ge 1$ such that ${\mathcal{O}}_X\otimes L^{\otimes d}\cong L^{\otimes d}$ is globally generated for every $d\ge d_1$.

The affine section-open cover adapted to the base. Assume $X\mathrm{\ne }\mathrm{\varnothing }$; the empty case is handled separately in the conclusion. Consider the family of all pairs $(V,a)$ where $V\subseteq S$ is affine open and $a\in \mathrm{\Gamma }(X,L^{\otimes n})$ is homogeneous of some degree $n\ge 1$, with $X_a$ affine and $X_a\subseteq f^{-1}(V)$. This family covers $X$: for each point $x$ there exists an affine open $V\subseteq S$ containing $f(x)$ by [F2], and step 3.1 applied to $f^{-1}(V)$ gives such an $a$ with $x\in X_a$. Quasi-compactness of $X$ from step 1.1 supplies finitely many pairs $(V_i,a_i)$, indexed by $i=0,\dots ,n$, whose nonvanishing loci cover $X$. Put $d_i=\mathrm{deg}(a_i)\ge 1$. This selects only finitely many witnesses; it does not choose a pair simultaneously for every point of $X$.

Global generation in the remaining degrees, uniformly in the future choice of $N$. For every integer $N\ge 1$ and every $d\ge d_1+N$, one has $d-N\ge d_1$, so $L^{\otimes (d-N)}$ is globally generated by step 3.2; since $X$ is quasi-compact by step 1.1 and the sheaf is invertible, finitely many sections $s_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,s_t^{\mathrm{\prime }\mathrm{\prime }}\in \mathrm{\Gamma }(X,L^{\otimes (d-N)})$ witness this global generation [F14].

The chart rings are finitely generated. For each $i$ the morphism $X_{a_i}\to V_i$ is locally of finite type, and both schemes are affine, so the ring map ${\mathcal{O}}_S(V_i)\to {\mathcal{O}}_X(X_{a_i})$ is of finite type [F2]; choose finitely many elements $f_{ij}\in {\mathcal{O}}_X(X_{a_i})$, $j=1,\dots ,n_i$, generating ${\mathcal{O}}_X(X_{a_i})$ as an ${\mathcal{O}}_S(V_i)$-algebra.

The second morphism for any pair $(N,d)$ from step 4.2. By [F7] the generating sections $s_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,s_t^{\mathrm{\prime }\mathrm{\prime }}$ determine an $S$-morphism $j^{\mathrm{\prime }}:X\to {\mathbb{P}}_S^{t-1}$ with $j^{\mathrm{\prime }\ast }\mathcal{O}(1)\cong L^{\otimes (d-N)}$.

Clearing denominators. For each $i,j$ the section $f_{ij}\in \mathrm{\Gamma }(X_{a_i},{\mathcal{O}}_X)$ lies over the nonvanishing locus of $a_i$, so [F5] applied to $X$ (quasi-compact and quasi-separated by steps 1.1 and 2.1), the quasi-coherent sheaf ${\mathcal{O}}_X$, the invertible sheaf $L$, $d_i>0$ and the section $a_i$ gives $r_{ij}\ge 0$ and $a_{ij}\in \mathrm{\Gamma }(X,L^{\otimes d_i{r}_{ij}})$ with $f_{ij}=a_{ij}\mathrm{/}{a}_i^{r_{ij}}$ on $X_{a_i}$; replacing $r_{ij}$ by $r_{ij}+1$ and $a_{ij}$ by $a_{ij}\otimes a_i$ if necessary we may assume $r_{ij}\ge 1$, so that $a_{ij}\in A_{+}$ is homogeneous of degree $d_{ij}=d_i{r}_{ij}$.

The common degree and the generating family. Choose $N\ge 1$ a common multiple of all degrees $d_i$ and $d_{ij}$ so large that $N\mathrm{/}{d}_i\ge r_{ij}$ for all $i,j$, and put $b_i=a_i^{N\mathrm{/}{d}_i}\in A_N$ and $b_{ij}=a_{ij}{a}_i^{N\mathrm{/}{d}_i-r_{ij}}\in A_N$. On $X_{a_i}$ one has $b_i=a_i^{N\mathrm{/}{d}_i}$ and $b_{ij}=f_{ij}{b}_i$ by step 6.1, so the section $b_i$ is nonvanishing exactly on $X_{a_i}$; since the $X_{a_i}$ cover $X$ by step 4.1, at every point of $X$ some member of the finite family $\{b_i,b_{ij}\}$ has nonzero value. Hence this family generates the invertible sheaf $L^{\otimes N}$ [F14].

The morphism $j$. By [F7] the generating family $\{b_i,b_{ij}\}$ of $L^{\otimes N}$ determines a unique $S$-morphism $j:X\to {\mathbb{P}}_S^m$, where $m=n+{\sum }_i{n}_i$, with $j^{\ast }\mathcal{O}(1)\cong L^{\otimes N}$; naming the coordinates $T_0,\dots ,T_n,T_{ij}$ one has $j^{-1}(D_{+}(T_i))=X_{b_i}=X_{a_i}$ and $(T_{\alpha }\mathrm{/}{T}_i)\circ j=b_{\alpha }\mathrm{/}{b}_i$ on $X_{a_i}$ for every coordinate index $\alpha \mathrm{\ne }i$.

The product morphism. Let $\{s_{\alpha }^{\mathrm{\prime }}\}$ denote the family $\{b_i,b_{ij}\}$ of step 7.1; the products $s_{\alpha }^{\mathrm{\prime }}{s}_{\beta }^{\mathrm{\prime }\mathrm{\prime }}\in \mathrm{\Gamma }(X,L^{\otimes d})$ generate $L^{\otimes d}$, because at every point some $s_{\alpha }^{\mathrm{\prime }}$ and some $s_{\beta }^{\mathrm{\prime }\mathrm{\prime }}$ are nonvanishing there and the tensor product of sections is nonvanishing at that point [F14]. Hence by [F7] they determine an $S$-morphism $i:X\to {\mathbb{P}}_S^{k-1}$, where $k-1=(m+1)t-1$, with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes d}$.

Chartwise closed immersions. Fix $i$ and write $V_i=\mathrm{Spec}{R}_i$ and $X_{a_i}=\mathrm{Spec}{B}_i$. The chart $D_{+}(T_i)\cap {\mathbb{P}}_{V_i}^m$ is the affine scheme $\mathrm{Spec}{R}_i[T_{\alpha }\mathrm{/}{T}_i:\alpha \mathrm{\ne }i]$ [F8], and by step 8.1 the restriction of $j$ to $X_{a_i}$ corresponds to the $R_i$-algebra map sending $T_{ij}\mathrm{/}{T}_i$ to $f_{ij}$ and $T_{i^{\mathrm{\prime }}}\mathrm{/}{T}_i$ to $b_{i^{\mathrm{\prime }}}\mathrm{/}{b}_i$. This map is surjective because the $f_{ij}$ generate $B_i$ over $R_i$ by step 5.1; hence the restriction $X_{a_i}\to D_{+}(T_i)\cap {\mathbb{P}}_{V_i}^m$ is a closed immersion [F9].

The composite with the Segre embedding. By [F13] the Segre embedding $\sigma :{\mathbb{P}}_S^m{\times }_S{\mathbb{P}}_S^{t-1}\to {\mathbb{P}}_S^{k-1}$ is a closed immersion with ${\sigma }^{\ast }\mathcal{O}(1)\cong p{r}_1^{\ast }\mathcal{O}(1)\otimes p{r}_2^{\ast }\mathcal{O}(1)$, and it carries the coordinate section $z_{\alpha \beta }$ to $p{r}_1^{\ast }{x}_{\alpha }\otimes p{r}_2^{\ast }{y}_{\beta }$. Therefore the composite $\sigma \circ (j,j^{\mathrm{\prime }})$ is an $S$-morphism with pullback $j^{\ast }\mathcal{O}(1)\otimes j^{\mathrm{\prime }\ast }\mathcal{O}(1)\cong L^{\otimes N}\otimes L^{\otimes (d-N)}\cong L^{\otimes d}$ of $\mathcal{O}(1)$, and it pulls the coordinate section $z_{\alpha \beta }$ back to $s_{\alpha }^{\mathrm{\prime }}{s}_{\beta }^{\mathrm{\prime }\mathrm{\prime }}$; since by [F7] a morphism to projective space with pullback $L^{\otimes d}$ and prescribed coordinate pullbacks is unique, $i=\sigma \circ (j,j^{\mathrm{\prime }})$.

The morphism $j$ is an immersion. Let $W={\bigcup }_i(D_{+}(T_i)\cap {\mathbb{P}}_{V_i}^m)$, an open subscheme of ${\mathbb{P}}_S^m$ containing $j(X)$ by step 8.1. The opens $j^{-1}(D_{+}(T_i)\cap {\mathbb{P}}_{V_i}^m)=X_{a_i}$ cover $X$ and each restriction $X_{a_i}\to D_{+}(T_i)\cap {\mathbb{P}}_{V_i}^m$ is a closed immersion by step 9.1, so locality of closed immersions on the target [F9] makes $j:X\to W$ a closed immersion; composing with the open immersion $W\hookrightarrow {\mathbb{P}}_S^m$ exhibits $j:X\to {\mathbb{P}}_S^m$ as an immersion [F10].

$(j,j^{\mathrm{\prime }})$ is an immersion. The graph ${\mathrm{\Gamma }}_{j^{\mathrm{\prime }}}=({\mathrm{id}}_X,j^{\mathrm{\prime }}):X\to X{\times }_S{\mathbb{P}}_S^{t-1}$ is the base change of the diagonal ${\mathrm{\Delta }}_{{\mathbb{P}}_S^{t-1}\mathrm{/}S}$ [F11]; the projection ${\mathbb{P}}_S^{t-1}\to S$ is proper, hence separated [F12], so that diagonal is a closed immersion and the graph is a closed immersion as a base change of one [F9]. The morphism $j\times \mathrm{id}:X{\times }_S{\mathbb{P}}_S^{t-1}\to {\mathbb{P}}_S^m{\times }_S{\mathbb{P}}_S^{t-1}$ is the base change of the immersion $j$ of step 10.1, hence an immersion [F10]. Since $(j,j^{\mathrm{\prime }})=(j\times \mathrm{id})\circ {\mathrm{\Gamma }}_{j^{\mathrm{\prime }}}$, step 1.3(b) shows that $(j,j^{\mathrm{\prime }})$ is an immersion.

The morphism $\sigma \circ u$ is an immersion. By step 11.1 and [F10] write $(j,j^{\mathrm{\prime }})=u\circ c$ with $c:X\to X^{\mathrm{\prime }}$ a closed immersion into an open subscheme $u:X^{\mathrm{\prime }}\hookrightarrow {\mathbb{P}}_S^m{\times }_S{\mathbb{P}}_S^{t-1}$. Put $\mathrm{\Omega }={\mathbb{P}}_S^{k-1}\setminus (\sigma ({\mathbb{P}}_S^m{\times }_S{\mathbb{P}}_S^{t-1})\setminus \sigma (X^{\mathrm{\prime }}))$, an open subscheme of ${\mathbb{P}}_S^{k-1}$: its complement is closed because $\sigma$ has closed image [F12] and $\sigma (X^{\mathrm{\prime }})$ is open in that image, and by construction $\mathrm{\Omega }\cap \sigma ({\mathbb{P}}_S^m{\times }_S{\mathbb{P}}_S^{t-1})=\sigma (X^{\mathrm{\prime }})$. The restriction ${\sigma }^{-1}(\mathrm{\Omega })\to \mathrm{\Omega }$ of $\sigma$ is the base change of the closed immersion $\sigma$ along the open immersion $\mathrm{\Omega }\hookrightarrow {\mathbb{P}}_S^{k-1}$, hence a closed immersion [F9], and ${\sigma }^{-1}(\mathrm{\Omega })=X^{\mathrm{\prime }}$ by step 11.1; therefore $X^{\mathrm{\prime }}\to \mathrm{\Omega }$ is a closed immersion and composing with the open immersion $\mathrm{\Omega }\hookrightarrow {\mathbb{P}}_S^{k-1}$ presents $\sigma \circ u$ as an immersion [F10].

$i$ is an immersion. By steps 9.2, 11.1 and 12.1 the morphism $i=\sigma \circ (j,j^{\mathrm{\prime }})=(\sigma \circ u)\circ c$ is the composite of the immersion $\sigma \circ u$ with the closed immersion $c$, hence an immersion by step 1.3(b).

$i$ is proper. The hypothesis makes $X\to S$ proper, and ${\mathbb{P}}_S^{k-1}\to S$ is proper [F12], hence separated, so the $S$-morphism $i$, being a morphism from a proper $S$-scheme to a separated $S$-scheme, is proper by the AC-qualified [F12].

$i$ is a closed immersion. A proper morphism is closed [F12], so the image $i(X)$ is closed in ${\mathbb{P}}_S^{k-1}$; an immersion with closed image is a closed immersion [F12].

Conclusion. The morphism $i$ is a quasi-compact $S$-immersion with $i^{\ast }\mathcal{O}(1)\cong L^{\otimes d}$: it is a closed immersion by step 15.1, and a closed immersion is of finite type, hence quasi-compact [F11]. Therefore $L^{\otimes d}$ is closed H-very ample relative to $S$ as defined in Relative very ampleness in the finite projective-space convention, and since, after choosing $N$ in step 7.1, the integer $d\ge d_0=d_1+N$ was arbitrary, this bound proves the theorem. The endpoint $d=d_0$ is included: it uses $L^{\otimes N}$ from step 8.1 and $L^{\otimes d_1}$ from step 4.2; the exponent $N\mathrm{/}{d}_i-r_{ij}$ in step 7.1 is $\ge 0$ by the choice of $N$ and may be zero, in which case $b_{ij}=a_{ij}$; the single-chart case $n=0$ and the case $m=0$ are allowed, the target being ${\mathbb{P}}_S^0\cong S$. If $X=\mathrm{\varnothing }$ then $L$ is ample vacuously and every $L^{\otimes d}$ is closed H-very ample via the empty morphism $\mathrm{\varnothing }\to {\mathbb{P}}_S^0\cong S$, which is a closed immersion with pullback of $\mathcal{O}(1)$ equal to the unique invertible sheaf of the empty scheme; the construction above is vacuous in this case and $d_0=1$ works. The recursive open-cover selection in [F17] uses [A1]; the other uses of choice are inherited through the suppliers of [F12] and the projective-space constructions. Besides these, only finitely many section opens, algebra generators and witnessing sections are selected. [A1, F11, F12, F17, step 7.1, step 8.1, step 4.2, step 15.1, cases: X empty and d endpoint] \qed