I believe I read somewhere that $\mathbb{Z}$-by-residually finite groups are residually finite. That is, if $G/N$ is residually finite where $N\cong \mathbb{Z}$ then $G$ is residually finite.

However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if someone could confirm whether this is true or not, and if it is give either a proof or a reference for this result? (If not, a counter-example would not go amiss!)

Note that I definitely know it is true if $G/N$ is f.g. free (there is a paper of B. Baumslag and Levin which mentions this, called "A class of one-relator groups with torsion", if I remember correctly).