Proof using Induction? Any thoughts on how to prove the following proposition
If $x_1>0$ and $a>0$ and $x_{ n+1 }=\frac { 1 }{ 2 } \left( x_{ n }+\frac { a^{ 2 } }{ x_{ n } }  \right) $ then $a\leq x_{n+1}\leq x_n$.

First, you see that $x_n > 0$ for all $n$.
Then $x_{n+1}=\dfrac{a}{2}(\dfrac{x_n}{a}+\dfrac{a}{x_n}) \geq \dfrac{a}{2} 2 =a$
And $x_{n+1}-x_n=\dfrac{a^2-x_n^2}{2x_n}$
And you have to assume that $x_1 > a$ to make the induction works.
