Can't understand proof of excision property

I was reading the proof here. I understood almost all the parts after Exercise 4, such as defining $bs_n ^X$ inductively, a chain homotopy $(R^X )$, and that we get a small chain $(bs_n ^X)^k$ by Baire Category (I guess). However, I can't understand that if $(bs_n ^X)^k (\alpha )$ is a small chain, and $R_n ^X (\alpha )$ is small, then inclusion of $C'(X) \rightarrow C(X)$ induces isomorphisms in homology. (Exercise 4) I get this intuitively, but it is not explicit.

-

This isn't a trivial step in proving excision property: in my personal opinion this is a tricky part of the proof and a little work is required to find a left inverse to the inclusion map $C_\bullet(X) \to C'_\bullet(X)$ which is also a right homotopy inverse (i.e. an inverse up to chain homotopy).
Actually Hatcher proves a little more general fact: for every family $\mathcal U = \{\mathcal U_i\}_{i}$ of subspace of $X$ such that $\bigcup_i \mathcal{\stackrel{\circ}U_i} = X$, the sub-chain complex of $C_\bullet(X)$ generated by all the singular simplexes which have image contained in one of the $\mathcal U_i$ is such the inclusion induce isomorphism in homology.