Let $J^\bullet$ be an acyclic complex of injective objects in an abelian category $\mathcal A$. Consider its finite subquotient complexes of canonical truncation $0\to Z^m\to J^m\to J^{m+1}\to \dotsb\to J^{n-1}\to Z^n\to 0$, where $Z^i$ denotes the kernel of the differential $J^i\to J^{i+1}$. This finite complex is a right resolution of the object $Z^m$ whose terms are all injective objects in $\mathcal A$, with a possible exception of the rightmost term $Z^n$. Now if the category $\mathcal A$ has finite homological dimension and the number $n-m$ is chosen to be large enough, it follows that the object $Z^n$ is also injective. Finally, an acyclic complex of injectives $J^\bullet$ with injective objects of cocycles $J^i$ is easily seen to be contractible.

Let $J^\bullet$ be an acyclic complex of injective objects in an abelian category $\mathcal A$. Consider its finite subquotient complexes of canonical truncation $0\to Z^m\to J^m\to J^{m+1}\to \dotsb\to J^{n-1}\to Z^n\to 0$, where $Z^i$ denotes the kernel of the differential $J^i\to J^{i+1}$. This finite complex is a right resolution of the object $Z^m$ whose terms are all injective objects in $\mathcal A$, with a possible exception of the rightmost term $Z^n$. Now if the category $\mathcal A$ has finite homological dimension and the number $n-m$ is chosen to be large enough, it follows that the object $Z^n$ is also injective. Finally, an acyclic complex of injectives $J^\bullet$ with injective objects of cocycles $Z^i$ is easily seen to be contractible.