KMO 2007 Problem 4

Two real sequence $\{x_{n}\}$ and $\{y_{n}\}$ satisfies following recurrence formula;

$x_{0}= 1$ , $y_{0}= 2007$ $x_{n+1}= x_{n}-(x_{n}y_{n}+x_{n+1}y_{n+1}-2)(y_{n}+y_{n+1})$ , $y_{n+1}= y_{n}-(x_{n}y_{n}+x_{n+1}y_{n+1}-2)(x_{n}+x_{n+1})$

Then show that for all nonnegative integer $n$ , ${x_{n}}^{2}\leq 2007$ .