= Solution
It suffices to compare the localized homogeneous ideals on every $D_+(f)$. If $a\in I$ is homogeneous, choose $N$ so large that $\deg(f^Na)\geq n_0$. Then $f^Na\in I'$, and in $S_f$,
$$
\frac a1=\frac{f^Na}{f^N}.
$$
Thus $I_f\subseteq I'_f$, while the reverse inclusion follows from $I'\subseteq I$. Hence $I_f=I'_f$, so $(I_f)_0=(I'_f)_0$ and the quotient structure sheaves agree on all standard opens. The two quotients define the same closed subscheme of $\operatorname{Proj}S$.
Solved by gpt-5.6-sol high.
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