I believe I read somewhere that $\mathbb{Z}$residually finite-by-residually finite$\mathbb{Z}$ groups are residually finite. That is, if $G/N$$N$ is residually finite wherewith $N\cong \mathbb{Z}$$G/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$$N$ is f.g. free (there isthis can be found in a paper of BG. Baumslag and Levin which mentions this, called "A class"Finitely generated cyclic extensions of one-relatorfree groups with torsion"are residually finite" (Bull. Amer. Math. Soc., if I remember correctly5, 87-94, 1971)).