An ample line bundle on a finite-type scheme gives a projective immersion

If $X$ is empty, its empty closed immersion into ${\mathbf{P}}_k^0$ proves the assertion. Otherwise, by quasi-compactness and [F1], choose finitely many affine opens $X_{s_i}$ covering $X$, with $s_i\in \mathrm{\Gamma }(X,L^{n_i})$ and $n_i>0$. Each has a finitely generated coordinate ring over $k$; choose generators $a_{ij}$. By [F1] there are positive $r_{ij}$ and sections $t_{ij}\in \mathrm{\Gamma }(X,L^{n_i{r}_{ij}})$ with $t_{ij}\mathrm{/}{s}_i^{r_{ij}}=a_{ij}$ on $X_{s_i}$. Choose a common multiple $N$ of the $n_i$, large enough that $N\mathrm{/}{n}_i\ge r_{ij}$ for all $i,j$. Put $b_i=s_i^{N\mathrm{/}{n}_i}$ and $b_{ij}=t_{ij}{s}_i^{N\mathrm{/}{n}_i-r_{ij}}$, all sections of $L^N$.

The $b_i$ have the same nonvanishing loci as the $s_i$, so these sections generate $L^N$. By [F2] the family $b_i,b_{ij}$ defines $j:X\to {\mathbf{P}}_k^M$. On the coordinate chart where $b_i\mathrm{\ne }0$, the ratios $b_{ij}\mathrm{/}{b}_i$ pull back to $a_{ij}$. The induced map from that affine projective-chart ring to $\mathrm{\Gamma }(X_{s_i},\mathcal{O})$ is therefore onto, hence $X_{s_i}$ is a closed subscheme of the chart. These charts cover an open $W\subset {\mathbf{P}}^M$ containing $j(X)$, and their inverse images cover $X$. Closed immersions are local on the target by the affine quotient description, so $X\to W$ is a closed immersion. Composing with $W\subset {\mathbf{P}}^M$ gives the required locally closed immersion. AC is inherited from [F1]–[F2].