Homology of infinite dimensional real projective space given by Tor-functor

Question

Let $Z$ be the trivial $Z[Z/2]$-module (i.e. $Z/2$ acts trivially). How can one show that for all $n\geq0$ $Tor_n^{Z[Z/2]}(Z,Z) = H_n (RP^{\infty},Z)$ without calculating Tor and the homology of the infinite dimensional real projective space explicitly?

Answer 1

Citing the answer in the comments by M. Miller:

Use a cell decomposition of projective space to give an invariant cell decomposition of $S^\infty$. The cellular chain complex will be a free resolution of $\Bbb Z$ as a $\Bbb Z[\Bbb Z/2]$ module.