Convergence of Ratio Test implies Convergence of the Root Test

By definition of limit, for each $\varepsilon>0$ there exists $N$ s.t. $$n>N \implies \left| \left| \frac{a_{n+1}}{a_n} \right|-L \right|<\varepsilon.$$ So $$|a_n|=\frac{|a_n|}{|a_{n-1}|}\cdots \frac{|a_{N+1}|}{|a_N|} |a_N|<(L+\varepsilon) ^{n-N} |a_N|$$ Take the $n$th root of both sides of the inequality. Then we get $$\sqrt[n]{|a_n|} <(L+\varepsilon)^{1-N/n}\sqrt[n]{|a_N|}.$$ Taking $n\to\infty$ then $$\lim_{n\to\infty}\sqrt[n]{|a_n|} \le L+\varepsilon.$$ Since $\varepsilon$ is arbitrary, we get $\lim_{n\to\infty}\sqrt[n]{|a_n|} \le L.$ Likewise we can get $\lim_{n\to\infty}\sqrt[n]{|a_n|} \ge L.$

This can also be proven by using Stolz theorem as shown in Fichtenholz's 'Differential and Integral Calculus'. We also need to know some facts about logarithms and exponentiation.

Stolz Theorem: Suppose that $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ are sequences of real numbers. Assume that $(a_n)_{n\geq 1}$ is a strictly increasing sequence, divergent to $\infty$, and that $$\lim_{n\to\infty} \frac{b_{n+1}-b_n}{a_{n+1}-a_n}\ \text{exists (possibly is infinite)}. $$ Then $\lim\limits_{n\to\infty} \frac{b_n}{a_n}$ exists (which might also be infinite) and $$\lim_{n\to\infty} \frac{b_{n+1}-b_n}{a_{n+1}-a_n} = \lim_{n\to\infty} \frac{b_n}{a_n}. $$

Let's assume that $(a_n)_{n\geq 1}$ is a sequence of positive numbers and $\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}$ exists. Then using Stolz theorem:

$$\lim_{n\to\infty}\ln(a_n^{1/n}) = \lim_{n\to\infty}\frac{\ln(a_n)}{n} = \lim_{n\to\infty} \big(\ln(a_{n+1})-\ln(a_n)\big) = \lim_{n\to\infty} \ln\left(\frac{a_{n+1}}{a_n}\right),$$ from which immediately $$\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \sqrt[n]{a_n}. $$

Note: For the case when $\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 0$ the proof is still valid, we still get that $\lim_{n\to\infty} \ln(\sqrt[n]{a_n}) = -\infty$, which can happen if and only if $\lim_{n\to\infty} \sqrt[n]{a_n} = 0$.