I am currently writing my thesis and looking for a reference (or a short proof) to the following fact:            

> Let $N$ be a finitely generated nilpotent group, and denote its central series by $N_r$, that is, $N_1=N$ and $N_{k+1}=[N,N_k]$ is the commutator group of $N$ amd $N_k$. Then there is a finite index subgroup $H$ of $N$ which has the following property - if $H_r$ is $H$'s central sequence, then all the quotients $\frac{H_r}{H_{r+1}}$ are torsion free.

This fact is quoted in "Random walks on infinite groups and graphs" by Wolfgang Woess during the classification of recurrent groups.

Thanks in advance to anyone who is willing to help :-).