Two real sequences $(x_n)$ and $(y_n)$ are defined by
$$x_{n+1}=x_n-(x_ny_n+x_{n+1}y_{n+1}-2)(y_n+y_{n+1})$$
$$y_{n+1}=y_n-(x_ny_n+x_{n+1}y_{n+1}-2)(x_n+x_{n+1})$$ with $x_0=1$ and $y_0=2007.$ I need to show that $|x_n|\lt \sqrt{2007}$ for all $n\in\mathbb{N}.$
I proved that $$x_{n+1}^2-x_n^2=y_{n+1}^2-y_n^2\,\,\,\,\,\,\,\,\,\,\forall n\in\mathbb{N},$$ which implies $|x_n|\lt|y_n|$ and $$x_n^2=y_n^2-2007^2+1.$$
Also I would like to know that, Is $x_n$ convergent? Any Idea?