I have been stuck trying to show $\limsup_{n\to\infty}s_n^{1/n}\leq\limsup_{n\to\infty}s_{n+1}/s_n$ for any sequence of positive reals, $s$ given that $\limsup_{n\to\infty}s_{n+1}/s_n=\lim_{n\to\infty}s_{n+1}/s_n=L\in\mathbb{R}$ I wonder if it would be appropriate to combine the limits:
$\lim_{n\to\infty}\left(\sup_{m>n}s_m^{1/n}-\sup_{m>n}s_{m+1}/s_m\right)\geq0$
Since we know so little about $s$, I don't think combining the $\sup$ will be very helpful. So how can we find a relationship between these to prove the inequality? Even if we were to combine them, we don't know any relationship between $s_n$ and $s_{n+1}$.
Of course, $s_{n+1}/s_n$ is bounded, so perhaps we can use this to but a bound on $\sup s_n^{1/n}$ and pin it between $0$ and $L$ instead of messing with combining the limits. This may be preferable since it seems impossible to establish a relationship between the elements of the two derived sequences, given so little information about $s$.
