The Stacks project

Proof. This follows from Cohomology on Sites, Lemma 21.16.1. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ be the set of quasi-compact and quasi-separated spaces étale over $X$. Note that if $U \in \mathcal{B}$ then, because $U$ is quasi-compact, the collection of finite coverings $\{ U_ i \to U\}$ with $U_ i \in \mathcal{B}$ is cofinal in the set of coverings of $U$ in $X_{spaces, {\acute{e}tale}}$. By Morphisms of Spaces, Lemma 67.8.10 the set $\mathcal{B}$ satisfies all the assumptions of Cohomology on Sites, Lemma 21.16.1. Since $X \in \mathcal{B}$ we win. $\square$