The Stacks project

Proof. Proof of (3). Observe that $\mathop{\mathrm{Spec}}(R) - V(I)$ has a covering by the affine opens $\mathop{\mathrm{Spec}}(R_ f)$ for $f \in I$. The rings $R_ f$ are Jacobson by More on Algebra, Lemma 15.10.5. Hence $\mathop{\mathrm{Spec}}(R) \setminus V(I)$ is Jacobson by Lemma 28.6.3. Parts (1) and (2) are special cases of (3).

Direct proof of case (1). Since $\mathop{\mathrm{Spec}}(R)$ is a Noetherian scheme, $S$ is a Noetherian scheme (Lemma 28.5.6). Hence $S$ is a sober, Noetherian topological space (use Schemes, Lemma 26.11.1). Assume $S$ is not Jacobson to get a contradiction. By Topology, Lemma 5.18.3 there exists some non-closed point $\xi \in S$ such that $\{ \xi \} $ is locally closed. This corresponds to a prime $\mathfrak p \subset R$ such that (1) there exists a prime $\mathfrak q$, $\mathfrak p \subset \mathfrak q \subset \mathfrak m$ with both inclusions strict, and (2) $\{ \mathfrak p\} $ is open in $\mathop{\mathrm{Spec}}(R/\mathfrak p)$. This is impossible by Algebra, Lemma 10.61.1. $\square$