Lemma 64.6.2. An object $E$ of $D(\mathcal{A})$ is contained in $D^+(\mathcal{A})$ if and only if $H^ i(E) =0 $ for all $i \ll 0$. Similar statements hold for $D^-$ and $D^ b$.

**Proof.**
Hint: use truncation functors. See Derived Categories, Lemma 13.11.5.
$\square$

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