Solution
= Solution
A <chain homotopy> from $g$ to $f$ consists of homomorphisms $s_n:C_n\to C'_{n+1}$ such that
$$
f_n-g_n=d'_{n+1}s_n+s_{n-1}d_n
$$
for every $n$. If $z\in C_n$ is a cycle, $d_nz=0$, so $f_n(z)-g_n(z)=d'_{n+1}s_n(z)$ is a boundary. Hence $f_n(z)$ and $g_n(z)$ represent the same <homology class>. Every <homology class> has a cycle representative, proving $\boxed{f_*=g_*}$ in every degree.