Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$

This may have been asked before, however I was unable to find any duplicate. This comes from pg. 52 of "Mathematical Analysis: An Introduction" by Browder. Problem 14:

If $(a_n)$ is a sequence in $\mathbb R$ and $a_n > 0$ for every $n$. Then show: $$ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$$

The middle inequality is clear. However I am having a hard time showing the ones on the left and right. (It seems like the approach should be similar for each). This is homework, so it'd be great if someone could give me a hint to get started on at least one of the inequalities.

Thanks.

Solution 1:

A somewhat detailed hint for the right hand side inequality:

Suppose $$ r := \lim \sup \frac{a_{n+1}}{a_n}. $$ (If the above expression is $\infty$, then there is nothing to prove. So assume $0 \leq r < \infty$.) Fix any $\epsilon > 0$. This means that there exists $N$ such that for $n \geq N$, we have $$ \frac{a_{n+1}}{a_n} \leq r + \epsilon. $$ From this, can you deduce that for $n \geq N$, we have $$ \frac{a_n}{a_N} \leq (r+\epsilon)^{n-N}? $$ Rearranging a bit, $$ a_n \leq (r+\epsilon)^{n} \left( \frac{a_N}{(r+\epsilon)^N} \right), $$ so that $$ a_n^{1/n} \leq (r+\epsilon) \left( \frac{a_N}{(r+\epsilon)^N} \right)^{1/n}. $$ Can you take it from here?

You do not have to work so hard for the left hand side inequality: there is a simple way to obtain the left hand side inequality using the right hand side one in a black-box way. I will leave you to figure it out.