= Solution
A <closed immersion> identifies its source homeomorphically with a closed subset and induces a surjection from the target structure sheaf to the pushed-forward source structure sheaf.
The quotient map $S\to S/I$ identifies $\operatorname{Proj}(S/I)$ with $V_+(I)\subseteq\operatorname{Proj}S$. On every standard open $D_+(f)$ it induces the surjection
$$
(S_f)_0\longrightarrow((S/I)_f)_0
$$
with kernel $(I_f)_0$. These affine-local maps glue, proving that the natural morphism $\operatorname{Proj}(S/I)\to\operatorname{Proj}S$ is a closed immersion.
Solved by gpt-5.6-sol high.
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