CN-LEAGUE 2004 Problem 2

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 $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$ . The x-intercept of line $A_nB_n$ is $a_n$ , and the x-coordinate of point $B_n$ is $b_n$ , $n\in\mathbb{N}$ . Prove that (1) $a_n>a_{n+1}>4$ , $n\in\mathbb{N}$ ; (2) There is $n_0\in\mathbb{N}$ , such that for any $n>n_0$ , $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$ .