Q:

How to show alternate term order of given sequence?

I am solving excercise of Baby Rudin Chapter 3 problem 17. Fix $\alpha >1$. Take $x_1>{\sqrt  \alpha }$.
and define 
$$x_{n+1}=\frac {\alpha+x_n}{1+x_n}$$
Then show that $x_1>x_3>x_5>x_7.....  $ 
 and $ x_2 < x_4<x_6<x_8<....$ 
I can show limit of sequence of $x_n$ as $ {\sqrt \alpha}$ by substituting $x$ instead of recurring term in given relation and solving relation. I had tried to solve relation that have to find but not able to find .Any help will be appreciated .Thanks a lot.

A:

Clearly $x_n>0$ for each $n$. Now calulate
\begin{eqnarray} d_n:= x_{n+2}-x_n &=& {\alpha +\frac {\alpha+x_n}{1+x_n}\over 1+\frac {\alpha+x_n}{1+x_n}}-x_n \\ &=&\frac {2\alpha+x_n(\alpha+1)}{1+\alpha+2x_n}-x_n\\&=& 2\frac {\alpha-x_n^2}{1+\alpha+2x_n}
\end{eqnarray}
So if $x_n>\sqrt{\alpha}$ then $d_n<0$, so $x_{n+2}<x_n$ and
if $x_n<\sqrt{\alpha}$ then $d_n>0$, so $x_{n+2}>x_n$

