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Possible Duplicate:
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.

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    This inequality appeared in [this answer](http://math.stackexchange.com/questions/28476/finding-the-limit-of-frac-n-sqrtnn/28487#28487) and [this answer](http://math.stackexchange.com/questions/76743/limit-of-fraca-n1a-n/76800#76800). See also [here](http://math.stackexchange.com/questions/69386/inequality-involving-limsup-and-liminf).2012-07-23
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    @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 $\limsup$.2012-07-23
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    @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](http://math.stackexchange.com/questions/tagged/limsup+inequality) to that question. (There are probably a few other questions which would deserve these tags, too.)2012-07-23
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    Anyway, if I wanted to search for similar questions, this Google search seems promising: [limsup frac site:math.stackexchange.com](http://www.google.com/#q=limsup+frac+site:math.stackexchange.com). Some tips and tricks for searching MSE can be found at meta in [this question](http://meta.math.stackexchange.com/questions/3258/how-do-you-search-for-duplicates) and other question tagged [search](http://meta.math.stackexchange.com/questions/tagged/search).2012-07-23
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    @MartinSleziak: Much appreciated!2012-07-23

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