This problem can be found in Kaczor, Nowak: Problems in Mathematical Analysis I, Real Numbers, Sequences and Series. I'll copy their solution here.
Problem 2.5.22, p.50, a solution is given on p.215.
Problem 2.5.22. The sequence $(a_n)$ is defined inductively as follows:
$$0<a_1<\pi \qquad a_{n+1}=\sin a_n \text{ for }n\ge 1$$
Prove that
$\lim\limits_{n\to\infty} \sqrt n a_n = \sqrt3$.
Solution:
It is easy to see that the sequence $(a_n)$ is monotonically
decreasing to zero. Moreover, an application of I'Hospital's rule gives
$$\lim\limits_{x\to 0}\frac{x^2-\sin^2x}{x^2\sin^2x}=\frac13.$$
Therefore
$$\lim\limits_{n\to\infty}\left(\frac1{a_{n+1}^2}-\frac1{a_n^2}\right)=\frac13$$
Now, by the result in Problem 2.3.14, $\lim\limits_{n\to\infty} na_n^2 = 3$.
Problem 2.3.14,
p.38, a solution is given on p.184.
Problem 2.3.14. Prove that if $(a_n)$ is a sequence for which
$$\lim\limits_{n\to\infty}(a_{n+1}-a_n)=a$$
then
$$\lim\limits_{n\to\infty}\frac{a_n}n=a.$$
Solution: In Stolz theorem we set $x_{n}=a_{n+1}$ and $y_n=n$.
Formulation of Stolz theorem in this book is the following
Let $(x_n)$, $(y_n)$ be two sequences that satisfy the conditions:
- $(y_n)$ strictly increases to $+\infty$,
- $$\lim\limits_{n\to\infty} \frac{x_n-x_{n-1}}{y_n-y_{n-1}}=g.$$
Then $$\lim\limits_{n\to\infty} \frac{x_n}{y_n}=g.$$
For Stolz-Cesaro theorem see also this question: Stolz-Cesàro Theorem
Perhaps it is also worth mentioning that there are two equivalent forms of Stolz-Cesaro theorem: see e.g. this answer.