Here is a question about infinitesimal things that I don't know how to prove

Question

Calculate the following limit

$$\displaystyle\lim_{n\to\infty}\sqrt[n]{(\sin1)^2+(\sin2)^2+...+(\sin n)^2}$$

Answer 1

We know that

$$ \sum_{k=1}^{n} \sin^2(k) = \frac{n}{2} + \frac{1-\csc(1) \sin(2n+1)}{4} $$ And since $|\sin(x)| \le 1$ we can say that $$ \frac{n}{2} + C_1 <\sum_{k=1}^{n} \sin^2(k) < \frac{n}{2} + C_2 $$ where $C_{1}, C_2$ are some constants. Can you conclude from here?