The Stacks project

Proof. The second assertion follows from the first (using Schemes, Lemma 26.12.4 and Lemma 28.5.6). Consider any nonempty affine open $U \subset X$. Let $x \in U$ be a closed point. If $x$ is a closed point of $X$ then we are done. If not, let $X_0 \subset X$ be the reduced induced closed subscheme structure on $\overline{\{ x\} }$. Then $U_0 = U \cap X_0$ is an affine open of $X_0$ by Schemes, Lemma 26.10.1 and $U_0 = \{ x\}$. Let $y \in X_0$, $y \not= x$ be a specialization of $x$. Consider the local ring $R = \mathcal{O}_{X_0, y}$. This is a Noetherian local ring as $X_0$ is Noetherian by Lemma 28.5.6. Denote $V \subset \mathop{\mathrm{Spec}}(R)$ the inverse image of $U_0$ in $\mathop{\mathrm{Spec}}(R)$ by the canonical morphism $\mathop{\mathrm{Spec}}(R) \to X_0$ (see Schemes, Section 26.13.) By construction $V$ is a singleton with unique point corresponding to $x$ (use Schemes, Lemma 26.13.2). By Algebra, Lemma 10.61.1 we see that $\dim (R) = 1$. In other words, we see that $y$ is an immediate specialization of $x$ (see Topology, Definition 5.20.1). In other words, any point $y \not= x$ such that $x \leadsto y$ is an immediate specialization of $x$. Clearly each of these points is a closed point as desired. $\square$