Is the following fact true?

Let $N$ be a finitely generated nilpotent group, and denote its lower central series by $(N_r)_{r\ge 1}$, that is, $N_1=N$ and $N_{k+1}=[N,N_k]$ is the commutator group of $N$ and $N_k$. 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.

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.

*Edit 3**(YC):* in the initial version of the question the fact was attributed to Woess in "Random walks on infinite groups and graphs" during the classification of recurrent groups, but actually Woess claims the weaker and more classical fact that every finitely generated nilpotent group has a torsion-free subgroup of finite index, for which a reference was given in one answer (after the 1st edit).

lowercentral series. $\endgroup$4more comments