Lemma 70.5.16. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of algebraic spaces over $Y$ with affine transition morphisms. Assume

$Y$ quasi-compact and quasi-separated,

$X_ i$ quasi-compact and quasi-separated,

the transition morphisms $X_{i'} \to X_ i$ are closed immersions,

$X_ i \to Y$ locally of finite type

$X \to Y$ is a closed immersion.

Then $X_ i \to Y$ is a closed immersion for $i$ large enough.

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