Properness and a relative ample line bundle give projectivity

Cover $S$ by finitely many affine opens. Relative ampleness says $A$ is ample over each such affine base. By [F1], choose powers giving closed projective-space embeddings on each open; take a common positive multiple of the finitely many exponents and use the Veronese monomials of the corresponding generating sections to obtain one power $A^k$ that gives a closed immersion on every open. Set $E=q_{\ast }{A}^k$, coherent by [F1]. The evaluation $q^{\ast }E\to A^k$ is onto on this cover, since its local sections include every member of these generating systems. It therefore defines $Y\to {\mathbb{P}}_S(E)$.

This map is a closed immersion. Locally choose one of the generating sections $s$ nonzero. The projective chart for $s$ has coordinate algebra generated by all ratios $t\mathrm{/}s$ with $t$ a local section of $E$. Already the finite ratios coming from the selected local projective embedding generate the coordinate algebra of $Y_s$ as a quotient; adding the other ratios preserves surjectivity onto that algebra. Thus on these charts the map is a closed immersion. They cover $Y$. The image is closed in the whole projective bundle because $Y\to S$ is proper and the projective bundle is separated over $S$; the graph is closed and its projection is a base change of the proper map. Hence the chart immersions give a global closed immersion. The last assertion is exactly the displayed extra global-section hypothesis.