Proof:

The first assertion follows from the comments preceding lemma (34.5). To show that the inclusion maps induce an injective map on homologies, let $\sigma \in S^{\;\cal U}_p(X)$ be a singular chain such that $\sigma = \partial \eta$ for some $\eta \in S_{p+1}(X)$. Choose $k \in \mathbb{N}$ such that ${\cal B}^k\eta \in S^{\;\cal U}_{p+1}(X)$. We have to show that ${\cal B}^k\eta$ is a boundary in $S^{\;{\cal U}}$. By exercise 5, ${\cal B}^k$ is chain homotopic to the identity via a homotopy $T_k$ say. Applying $\partial$ to

$${\cal B}^k\eta - \eta = T_k\partial \eta + \partial T_k\eta,$$

we see that $\partial ({\cal B}^k\eta) - \sigma = \partial T_k\sigma$. By (ii) of lemma (34.5), $\partial T_k\sigma \in S^{\;\cal U}_p(X)$ which means $\sigma$ is a boundary in $S_p^{\;{\cal U}}(X)$. To prove surjectivity, let $\sigma$ be a cycle in $S(X)$ and $k \in \mathbb{N}$ be such that ${\cal B}^k\sigma \in S^{\;\cal U}(X)$. From ${\cal B}^k\sigma - \sigma = \partial T_k\sigma$ we conclude that $\sigma$ is homologous to the cycle ${\cal B}^k\sigma$ in $S^{\;\cal U}(X)$. $\square$