Question 161
In a planar rectangular coordinate system, a sequence of points $\{A_n\}$ on the positive half of the $y$-axis and a sequence of points $\{B_n\}$ on the curve $y = \sqrt{2x}$ ($x \ge 0$) satisfy the condition $|OA_n| = |OB_n| = \frac{1}{n}$. The $x$-intercept of line segment $A_nB_n$ is $a_n$, and the $x$-coordinate of point $B_n$ is $b_n$, $n \in \mathbb{N}$. Prove that
(1) $a_n \gt a_{n+1} \gt 4$, $n \in \mathbb{N}$;
(2) There is $n_0 \in \mathbb{N}$, such that for any $n \gt n_0$, $\frac{b_2}{b_1} + \frac{b_3}{b_2} + \cdots + \frac{b_{n_0}}{b_{n_0-1}} + \frac{b_{n+1}}{b_n} \lt n - 2004$.
(1) $a_n \gt a_{n+1} \gt 4$, $n \in \mathbb{N}$;
(2) There is $n_0 \in \mathbb{N}$, such that for any $n \gt n_0$, $\frac{b_2}{b_1} + \frac{b_3}{b_2} + \cdots + \frac{b_{n_0}}{b_{n_0-1}} + \frac{b_{n+1}}{b_n} \lt n - 2004$.