Lemma 20.47.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet $ be a (locally) bounded above complex of $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $(m - i)$-pseudo-coherent for all $i$. Then $\mathcal{F}^\bullet $ is $m$-pseudo-coherent.

**Proof.**
Omitted. Hint: use Lemma 20.47.4 and truncations as in the proof of More on Algebra, Lemma 15.64.9.
$\square$

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