Fix $\displaystyle \alpha>1$, take $\displaystyle x_1 > \sqrt{\alpha}$ and define

$\displaystyle x_{n+1} = \frac{\alpha+x_n}{1+x_n} = x_n + \frac{\alpha-x_n^2}{1+x_n}$

I know this sequence should alternate about $\displaystyle \sqrt{\alpha}$. I.E., the odd terms are greater than $\displaystyle \sqrt{\alpha}$ and the even terms are less than $\displaystyle \sqrt{\alpha}$. However, I cannot seem to show this.

I've tried to use the fact that $\displaystyle \alpha>1$ and $\displaystyle \sqrt{\alpha} < x_n$ but I can't seem to get anything to work.

I would appreciate a hint very much.