# Residually finite nilpotent group

It is known that every finitely generated nilpotent group is residually finite. Why finitely generated hypothesis is essential?

It is essential in the proof, that the group is finitely generated. A finitely generated nilpotent group is supersolvable, and every supersolvable group is residually finite( a result of K.A. Hirsch). For the first part, the proof uses that the $k$-th group $G_k$ in the lower central series of the group $G$ is finitely generated for each $k$, so that each $G_k/G_{k+1}$ is finitely generated Abelian, i.e., a direct product of finitely many cyclic groups. This is not true for nilpotent groups which are not finitely-generated.
Edit: A possible counterexample is the additive group of $\mathbb{Q}$. It is abelian, hence nilpotent, but not residually finite.