Let $C_\bullet = (C_n, d_n)_{n \in \mathbb{Z}}$ be a chain complex of abelian groups, with boundary homomorphisms $d_n: C_n \to C_{n-1}$ satisfying $d_n \circ d_{n+1} = 0$ for every $n \in \mathbb{Z}$. Suppose $C_\bullet$ is contractible, meaning that there exist group homomorphisms $s_n: C_n \to C_{n+1}$ such that