Let $S_\bullet$ be a $(\mathbb{Z}_{\geq 0})$-graded ring, $f \in S_+$ be a homogenous element, $I \subseteq S_+$ any homogenous ideal, $V_+(I) := \{p \in ProjS_\bullet | I \subseteq p \}$.
I'm working through the text from Ravi Vakil on algebraic geometry and there is the following exercise:
Show that $V_+(I) \subseteq V_+(f)$ iff. $f \in \sqrt{I}$.
There is a hint stating, that one should "mimic" the proof of this statement for the affine case.
In the case of $Spec$ I just showed that $\sqrt I = \cap_{p \in V(I)}{p}$ - and it seems to me that this is the way Vakil had in mind, as in a hint for the affine case of the statement he referred to an earlier exercise showing $\sqrt I = \cap_{p\in V(I)}{p}$. Therefore I tried to prove $\sqrt I = \cap_{p \in V_+(I)}{p}$ but I'm only able to show '$\subseteq$' and I feel like '$\supseteq$' may be wrong in general.
So here is my question:
Is it true that $\sqrt I = \cap_{p \in V_+(I)}{p}$, if so - why is that. If it is not true, what do you think is the "intended" proof one should mimic?
I think I found a solution to the problem, but it is not really a mimic of my proof for $Spec$ and I'd really like to see that.