The Stacks project

Proof. Choose an affine scheme $U$ and an étale morphism $U \to X$ such that $x$ is in the image. Let $u_ i$, $i \in I$ be the generic points of irreducible components of $U$. Then each $u_ i$ maps to the generic point of $X$ (Decent Spaces, Lemma 68.20.1). By our definition of a decent space (Decent Spaces, Definition 68.6.1), we see that $I$ is finite. Hence $U = \mathop{\mathrm{Spec}}(A)$ where $A$ is a normal ring with finitely many minimal primes. Thus $A = \prod _{i \in I} A_ i$ is a product of normal domains by Algebra, Lemma 10.37.16. Then $U = \coprod U_ i$ with $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $x$ is in the image of $U_ i \to X$ for some $i$. This proves the lemma. $\square$