The Stacks project

Proof. Let $K^\bullet$ be a bounded above complex representing $K$, say $K^ i = 0$ for $i \geq n + 1$. After replacing $K^\bullet$ by $\tau _{\geq k + 1}K^\bullet$ we may assume $K^ i = 0$ for $i \leq k$. Then we may use the distinguished triangle

to see it suffices to prove the lemma for $K^ n[-n]$ and $\sigma _{\leq n - 1}K^\bullet$. By induction on $n$, we conclude that it suffices to prove the lemma in case $K$ is represented by the complex $M[-m]$ for some $R$-module $M$ and some $m \geq k + 1$. Since $R$ has global dimension $d$ by Algebra, Lemma 10.110.8 we see that $M$ has a projective resolution $0 \to P_ d \to \ldots \to P_0 \to M \to 0$. Then the complex $P^\bullet$ having $P_ i$ in degree $m - i$ is a bounded complex of projectives representing $M[-m]$. On the other hand, we can choose a complex $L^\bullet$ representing $L$ with $L^ i = 0$ for $i \geq k - d + 1$. Hence any map of complexes $P^\bullet \to L^\bullet$ is zero. This implies the lemma by Derived Categories, Lemma 13.19.8. $\square$