设数列 $\left\{x_n\right\},\left\{a_n\right\},\left\{b_n\right\}$ 分别满足 $x_n=\left(1+\sin \frac{1}{n}\right)^n, a_n=\frac{x_{2 n}}{x_{2 n-1}}, b_n=\prod_{i=1}^n a_i$.
(I) 求 $\lim _{n \rightarrow \infty} x_n$;
(II ) 证明: $\lim _{n \rightarrow \infty} b_n$ 存在.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$