Finite residual contains surjective endomorphism kernels ($\ker \theta \subseteq R_G$)

Let $G$ be a finitely generated group and $\theta :G\to G$ a surjective endomorphism. Pullback by $\theta$ permutes the finite set of normal subgroups of any prescribed finite index. For each such $N$, some iterate satisfies ${\theta }^{-m}(N)=N$, so $\ker \theta \subseteq \ker {\theta }^m\subseteq N$. Intersecting proves the inclusion. Consequently every finitely generated residually finite group is a Hopfian group.