Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 25I / c / Solution

Take the ring torus with minor radius $1$ and major radius $R_n=n+2$. With angular coordinate $\theta$ around the generating circle, its Gaussian curvature isand henceThus the displayed $\mathrm{limsup}$ condition holds, while every $S_n$ has genus one and is not diffeomorphic to the sphere.