= Solution
Because $\mathcal A_\bullet$ is coherent and locally generated by $\mathcal A_1$, there is locally a finite-rank coherent sheaf $\mathcal E$ and a surjection of graded algebras
$$
\operatorname{Sym}_{\mathcal O_X}(\mathcal E)\twoheadrightarrow\mathcal A_\bullet.
$$
The <Relative Proj construction> turns this into a <closed immersion>
$$
Y=\operatorname{Proj}_X\mathcal A_\bullet\hookrightarrow\mathbb P_X(\mathcal E).
$$
Consequently $Y\to X$ is a <projective morphism> and therefore a <proper morphism>. Since $X\to\operatorname{Spec}k$ is proper and proper morphisms are closed under composition, $Y\to\operatorname{Spec}k$ is proper.
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