Let $\{a_n\}$ be a sequence of real positive numbers. We wish to prove that $$ \lim \inf \frac{a_{n+1}}{a_n} \leq \lim \inf a_n^{1/n} \leq \lim \sup a_n^{1/n} \leq \lim \sup \frac{a_{n+1}}{a_n}$$
The answer to this post shows that $\lim \sup a_n^{1/n} \leq \lim \sup \frac{a_{n+1}}{a_n}$, and it mentions there is a black-box way to obtain $\lim \inf \frac{a_{n+1}}{a_n} \leq \lim \inf a_n^{1/n}$ from the previous inequality. What is it?
Thanks for your help in advance; I have no clue what this black-box method may be.
