Let $f(x)=1-\sin x$ for $x \in \mathbb{R}$. Define
\[ a_{n}=\sqrt[n]{f\left(\frac{1}{n}\right) f\left(\frac{2}{n}\right) \ldots f\left(\frac{n}{n}\right)}. \]
Then
- $\left\{a_{n}\right\}_{n}$ converges to $0$
- $\left\{a_{n}\right\}_{n}$ diverges to $\infty$
- $\left\{a_{n}\right\}_{n}$ converges and $\displaystyle{}\lim _{n \rightarrow \infty} a_{n}>0$
- none of the other three options is correct