Proof:

We show that $\delta_n = 0$ for every $n$ which would give us the sequences (37.7). For $c \in Z_n(X, A)$ we have the chains $r_{\sharp}(c) \in S_n(A)$ and $\partial _nc\in S_{n-1}(A)$. Now,

$$\partial _nr_{\sharp}(c) = r_{\sharp}(\partial _nc) = (r_{\sharp}... ... i_{\sharp})(\partial _nc) = (r\circ i)_{\sharp}(\partial _nc) = \partial _nc$$

Hence $\partial _nc$ is the boundary of the chain $r_{\sharp}(c) \in S_n(A)$ and so represents the zero element in $H_{n-1}(A)$. From lemma (37.3) we conclude that $\delta_n$ is the zero map. The short exact sequence (37.7) splits on the left since $H_n(r)\circ H_n(i)$ is the identity map on $H_n(A)$. $\square$