Solution
= 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> is
$$
K_n(\theta)=\frac{\cos\theta}{R_n+\cos\theta},
$$
and hence
$$
\inf_{S_n}K_n=-\frac1{R_n-1}\longrightarrow0.
$$
\b[Thus the displayed $\limsup$ condition holds, while every $S_n$ has genus one and is not diffeomorphic to the sphere.]