Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$

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Inequality involving $\limsup$ and $\liminf$
limit of $\frac{a_{n+1}}{a_n}$

Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$ and similarly $\liminf|s_n|^{1\over n}\ge \liminf|{s_{n+1}\over s_n}|$. I have no idea where to start. I tried to show the inequality through subsequence but still don't quite get where to start. Any explaination how to link to the concept of related topic would be appreciated.

This inequality appeared in this answer and this answer. See also here. — Martin Sleziak, Jul 23 '12 at 05:34
@MartinSleziak: How did you find those answers? Good memory or good search technique (or both, of course)? I have had limited success (read basically none) with searches for common terms such as $\limsup$. — copper.hat, Jul 23 '12 at 06:05
@copper.hat When I first saw this inequality at MSE, it was new to me and it seemed interesting, so I noted down the link. Later I added a few other links. Perhaps the next time searching will be easier, since I've added tags limsup+inequality to that question. (There are probably a few other questions which would deserve these tags, too.) — Martin Sleziak, Jul 23 '12 at 06:32
Anyway, if I wanted to search for similar questions, this Google search seems promising: limsup frac site:math.stackexchange.com. Some tips and tricks for searching MSE can be found at meta in this question and other question tagged search. — Martin Sleziak, Jul 23 '12 at 06:37

score 1 · Answer 1 · answered Jul 23 '12 at 05:46

Here’s a start for you.

For convenience we may suppose that the $s_n$ are non-negative and omit the absolute value signs. Suppose that $$\limsup_ns_n^{1\over n}>\limsup_n\,{s_{n+1}\over s_n}\;.$$ Let $$b=\frac12\left(\limsup_ns_n^{1\over n}+\limsup_n\,{s_{n+1}\over s_n}\right)\;;$$ then there is an $n_0\in\Bbb N$ such that $$\frac{s_{n+1}}{s_n}<b$$ for all $n\ge n_0$. Thus, for all $n>n_0$ we have $s_n<s_{n_0}b^{n-n_0}$. Let $a=s_{n_0}b^{-n_0}$, so that $s_n<ab^n$ for $n>n_0$. Then $s_n^{1/n}<a^{1/n}b$ for $n>n_0$.

What is $\lim\limits_{n\to\infty}a^{1/n}$?
How does $b$ compare with $\limsup\limits_n\,s_n^{1/n}$?

Show that $\limsup|s_n|^{1\over n}\le \limsup|{s_{n+1}\over s_n}|$

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