I'm reviewing my analysis, and have found the following problem on an old exam. I don't know how to solve it, so I'm hoping that, if someone can explain it to me, I can gain some familiarity with a technique that I must be overlooking.
I'm also very happy for any hints anyone may drop in the comments!
Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence taking values in the positive real numbers. Prove $$\limsup a_n^{1/n}\leq\limsup \frac{a_n}{a_{n-1}}$$
My main idea is to try to use the inequality $\limsup a_nb_n\leq (\limsup a_n)(\limsup b_n)$ to manipulate this into something nicer, but I can't quite see how to do that, and this may be altogether the wrong approach.
