Proving $\limsup_{n\to\infty}s_n^{1/n}\leq\limsup_{n\to\infty}s_{n+1}/s_n$
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$