I am currently reading a book on the Baum-Connes Conjecture by Mislin and Valette. And I have trouble understanding some arguments made on page 6 and 7. The condition necessary for the Whitehead theorem in 2.2 needs to be true for all subgroups, not just finite ones. One page later in the proof of 2.4 where he looks at the projection $$ pr_X: X \times \underline{E}G\longrightarrow X$$
it only needs to be true for finite subgroups. What am I missing?
Thank you
Theorem 2.4 assumes $X$ is proper, meaning that it is a $G$-CW complex whose point stabilizers are always finite subgroups in G. In other words, given an infinite subgroup $H$ of $G$, the fixed point space $X^H$ is empty. Thus the restriction of $pr_X$ on $H$-fixed points is just $\emptyset \rightarrow \emptyset$, a fortiori a homotopy equivalence.