Lemma 13.34.3. Let $\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $(K_ n)$ be an inverse system of $D^+(\mathcal{A})$. Then $R\mathop{\mathrm{lim}}\nolimits K_ n$ exists.

**Proof.**
It suffices to show that $\prod K_ n$ exists in $D(\mathcal{A})$. For every $n$ we can represent $K_ n$ by a bounded below complex $I_ n^\bullet $ of injectives (Lemma 13.18.3). Then $\prod K_ n$ is represented by $\prod I_ n^\bullet $, see Lemma 13.31.5.
$\square$

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