Problem
Suppose $0\to K'\xrightarrow iK\xrightarrow pK''\to 0$ is an exact sequence of chain complexes of modules over $R$, say. If chain maps $i,p$ are null-homotopic, then $K$ is contractible.
Discussion
It's an exercise in Dold's Lectures on Algebraic Topology, II.4(b), but complexes are assumed to be over abelian groups. I need a result of stronger form, since I need to show that if $f\colon B\to C$ is a chain map which has a homotopical inverse, then the mapping cone of $f$ is contractible, by analyzing the exact sequence $0\to C\to\operatorname{cone}(f)\to B[-1]\to 0$.
The later problem could be independently solved, cf. Spanier, Algebraic Topology, pp 167, Chapter 4, Sec 3, Theorem 10, but with an ingenious guesswork of the homotopy equivalence.
The problem is starred in Dold, so it might be hard. It's natural to choose the homotopy equivalence $s$ so that $si=s'$ where $s'\colon i\simeq0$ is null-homotopy. However, I have no idea how to take advantage of $p$. There's no obvious lifting of the null-homotopy of $p$.
Any idea? Thanks!