Could someone make the proof into a hinted exercise?

Question

Let $(c_n)$ be a sequence of positive numbers.

Could someone make the proof of the inequality

$\displaystyle\limsup_{n\to\infty}\sqrt[n]{c_n}\leq\limsup_{n\to\infty}\frac{c_{n+1}}{c_n}$

into a hinted exercise?

Answer 1

Fill-in the gaps:

Let $s:=\limsup\limits_{n\to\infty}\dfrac{c_{n+1}}{c_n}.$ There are two possibilities, either $s\in\mathbb R$ or $s=+\infty$ (why?) and since the latter is trivial, we will only consider the former.

Let $\varepsilon>0$ be given and let $N:=s+\varepsilon.$ It follows that there is some natural $M$ such that $\dfrac{c_{n+1}}{c_n}<N$ for all $n\geqslant M$ (why?) and since $c_i>0$ for each $i,$ we have (why?) $c_n<N^{n-M}c_M$ for all $n>M.$ Then (why?) $\limsup\limits_{n\to\infty}\sqrt[n]{c_n}\leqslant s.$