The Lemma is the following
Let $g:C\longrightarrow D$ be a cofibration and an $\widetilde{E}_*$-equivalence. Let $x\in D$ be a simplex. Then there is a subcomplex $D_0\subseteq D$ such that $X\in D_0,\ \#D_0\leq \beta$ and $$C\cap D_0\longrightarrow D_0$$ is an $\widetilde{E}_*$ equivalence
The solution is by defining a sequence of simplicial sets $$K_0\subseteq K_1 \subseteq K_2 \subseteq \ldots \subseteq D_0$$ and then setting $D_0=\bigcup K_n.$ For $K_0$, we choose any subsimplicial set with $\#K_0$ finite and $x\in K_0$.To produce $K_1$, for each homology class $y\in\widetilde{E}_*(K_0/K_0\cap C )$ there is a finite sub-complex $Z_y\subseteq D/C$ that maps to zero in $\widetilde{E}_*((K_0/K_0\cap C)\cup Z_y ).$
I dont quite understand why we can do that. Can somebody explain?