Vanishing ideal of a subset of $\operatorname{Proj} S_\bullet$

Question

I am reading FOAG by Vakil. In 4.5.K, I am asked to define $I(Z)$ where $Z\subseteq\operatorname{Proj}S_\bullet$ and $S_\bullet$ is a graded ring. The immediate idea of mine is to define it as $$I(Z):=\bigcap_{p\in Z}\mathfrak{p}$$ Now we see that it is indeed an homogeneous ideal as intersection of homogeneous ideals. Now we are left to show that it is prime and $I(Z)\subseteq S_+$. Here my attempts and doubts:
$\bullet$ Primeness of $I(Z)$: let $a\in S_d,b\in S_e$ be two homogeneous elements such that $ab\in I(Z)$. Then we have $ab\in \mathfrak{p}$ for all $\mathfrak{p}\in Z$. Now since $\mathfrak{p}$ is also homogeneous and prime, we conclude that either $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$. However, I cannot see how we can prove $a\in I(Z)$ or $b\in I(Z)$.
$\bullet \ I(Z)\subseteq S_+$: I am not so sure about this either. Because if we assume $I(Z)\nsubseteq S_+$, then there exists some $f\in I(Z)$ such that $f\notin S_+$. Then what can we see? I guess $f$ would contain some $0\neq f_0\in S_0$. Then we have $f_0\in\mathfrak{p}$ for all $\mathfrak{p}\in Z$. But I do not see where the contradiction appears.
Any help is appreciated! Thank you!