Let $S = \sum_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $> 0$. Let $D_+(f) = \{\mathfrak{p} \in\operatorname{Proj} S\mid f \notin \mathfrak{p}\}$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. The proposition states that
$D_+(f)$ is isomorphic to Spec $S_{(f)}$ as locally ringed spaces.
Part of his proof is as follows. For $\mathfrak{p} \in D_+(f)$, let $\psi(\mathfrak{p})=\mathfrak{p}S_f\cap S_{(f)}$. Then $\psi(\mathfrak{p}) \in$ Spec $S_{(f)}$. He wrote that the properties of localization show that $\psi\colon D_+(f) \rightarrow$ Spec $S_{(f)}$ is bijective. I wonder why $\psi$ is surjective.
This is taken from Iitaka's "Algebraic Geometry", Lemma 3.1(3):
Let $d=\text{deg}(f)$. Given $\mathfrak{p}\in\text{Spec}\left(S_{(f)}\right)$, we define $I_m$ to be $\{b\in S_m\mid b^d/f^m\in \mathfrak{p}\}$. It can be checked that $I_m$ is an abelian group, that $I:=\oplus I_m$ is a homogeneous prime ideal of $S$ not containing $f$, and that $\psi(I)=\mathfrak{p}$.