Hartshorne, Algebraic Geometry, Theorem III.5.2, reads (in part)
Theorem 5.2
Let $X$ be a projective scheme over a Noetherian ring, and let $\mathcal{O}_X(1)$ be a very ample invertible sheaf on $X$ over $\operatorname{Spec} A$. Let $\mathscr{F}$ be a coherent sheaf on $X$. Then:
(1) for each $i \geq 0$, $H^u(X, \mathscr{F})$ is a finitely-generated $A$-module;
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The proof proceeds by reducing to the case $X = \mathbb{P}^r_A$ and then checking things for $\mathcal{O}_X(q)$, which is fine. Then we need to establish things for arbitrary coherent sheaves. The proof here reads:
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In general, given a coherent sheaf $\mathscr{F}$ on $X$, we can write $\mathscr{F}$ as a quotient of a sheaf $\mathscr{E}$, which is a finite direct sum of sheaves $\mathcal{O}(q_i)$ for various integers $q_i$. Let $\mathscr{R}$ be the kernel, $$0 \to \mathscr{R} \to \mathscr{E} \to \mathscr{F} \to 0,$$ Then $\mathscr{R}$ is also coherent. We get an exact sequence of $A$-modules $$\cdots \to H^i(X, \mathscr{E}) \to H^i(X, \mathscr{F}) \to H^{i+1}(X, \mathscr{R}) \to \cdots$$ Now the module on the left is finitely generated because $\mathscr{E}$ is a sum of $\mathcal{O}(q_i)$, as remarked above. The module on the right is finitely generated by the induction hypothesis.
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This last sentence I don't understand. It doesn't explicitly state what the inductive hypothesis is, but I assume that it's that $H^j(X, \mathscr{F})$ is a finitely-generated $A$-module for $j > i$. In this case, what we know is that we have an exact sequence $$\cdots \to H^i(X, \mathscr{F}) \to H^{i+1}(X, \mathscr{R}) \to H^{i+1}(X, \mathscr{E}) \to H^{i+1}(X, \mathscr{F}) \to \cdots$$ where the two rightmost terms are finitely generated as $A$-modules. Since we don't seem to know anything about $H^i(X, \mathscr{F})$ on the left side yet, I don't see how this helps us establish anything about $H^{i+1}(X, \mathscr{R})$.
What have I missed?