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 lower 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 lower 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 :-).

**Edit:**
I read Woess' paper wrong - he actually only claims that $H$ has to be torsion free. Thanks for all the helpers.

***Edit 2:***
 Well, i'm still interested in an answer to the original question. I'll take off the reference request and make it a question instead.