Proposition 33.42.7. Let $X$ be a separated scheme such that every quasi-compact open has a finite number of irreducible components. Let $x_1, \ldots , x_ r \in X$ be points such that $\mathcal{O}_{X, x_ i}$ is Noetherian of dimension $\leq 1$. Then there exists an affine open subscheme of $X$ containing all of $x_1, \ldots , x_ r$.
Proof. We can replace $X$ by a quasi-compact open containing $x_1, \ldots , x_ r$ hence we may assume that $X$ has finitely many irreducible components. By Lemma 33.42.6 we reduce to the case where $X$ is integral. This case is Lemma 33.42.4. $\square$
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