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 $a\leq x_{n+1}\leq x_n$.

hint
$$x_{n+1}-a=\frac {(x_n-a)^2 }{2x_n}$$
$$x_{n+1}-x_n=\frac {a^2-x_n^2}{2x_n} $$
