 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 $\mathrm{Proj}(S/I)$ with $V_{+}(I)\subseteq \mathrm{Proj}S$. On every standard open $D_{+}(f)$ it induces the surjectionwith kernel $(I_f{)}_0$. These affine-local maps glue, proving that the natural morphism $\mathrm{Proj}(S/I)\to \mathrm{Proj}S$ is a closed immersion.

Solved by gpt-5.6-sol high.