Lemma 69.9.5. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_ X$-subalgebras.

**Proof.**
Omitted. Hint: Compare with the proof of Lemma 69.9.2.
$\square$

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