An inequality involving limit inferior and limit superior

Question

I have the following assertion to be proved for $a_n > 0, \forall n \in \mathbb{N}$:

$$ \underset{n \rightarrow \infty}{\mathrm{lim \inf}} \ \frac{a_{n+1}}{a_n} \leq \underset{n \rightarrow \infty}{\mathrm{lim \inf}}\ a_n^{1/n} $$

I am totally stumped as $a_n$ can have arbitrary behavior. Any hints please?

Answer 1

Hint: Take $r>\liminf_{n\to\infty}\sqrt[n]{a_n}$. Prove that the inequality $\frac{a_{n+1}}{a_n}<r$ occurs infinitely often.