The Stacks project

Proof. Since $X$ is quasi-compact, there exists $n$ and pairs $(\mathcal{L}_ i, s_ i)$, $i = 1, \ldots , n$ where $\mathcal{L}_ i$ is an invertible $\mathcal{O}_ X$-module and $s_ i \in \Gamma (X, \mathcal{L}_ i)$ is a section such that the set of points $U_ i \subset X$ where $s_ i$ is nonvanishing is affine and $X = U_1 \cup \ldots \cup U_ n$. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the image of $s_ i : \mathcal{L}_ i^{\otimes -1} \to \mathcal{O}_ X$. Applying Lemma 36.36.6 we find that $X$ has the resolution property. $\square$