# To show limit of $\left|\frac{ a_{n+1}}{a_n}\right|$ is smaller than lim inf $\left|(a_n)^{1/n}\right|$

Suppose that $a_{n}$ is a sequence of real numbers such that $\lim\limits_{n\to\infty} {\frac{\left|a_{n+1}\right|}{\left|(a_n)\right|}}$ exists, then $$\lim\limits_{n\to\infty} {\frac{\left|a_{n+1}\right|}{\left|a_{n}\right|}} \leq \liminf\limits_{n\to\infty} \left|a_n\right|^{1/n}$$.

This is how I have started:

Suppose $\lim\limits_{n\to\infty} {\frac{\left|a_{n+1}\right|}{\left|a_{n}\right|}}=L, L \in R$

Given any $\epsilon \gt 0$, there exist $n_0$ such that for all $n \ge n_0$, $$L - \epsilon \lt {\frac{\left|a_{n+1}\right|}{\left|a_{n}\right|}} \lt L+\epsilon$$

and after this I am quite clueless how to continue. Any help or hint is much appreciated. Please help if you could. Thank you very much.

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It may be helpful to see the relationships between the ratio test and root test lim infs and lim sups written out all at once. For this, see e.g. Proposition 11.29 of math.uga.edu/~pete/2400full.pdf –  Pete L. Clark Apr 10 at 15:15
@PeteL.Clark, can you please comment on my proof below? –  user70532 Apr 10 at 15:16