$consider \ the \ the \ sequence :$
$x_1=a (a \in R ) \ , \ x_{n+1}= \frac{1}{4}(1+x_n) , n \geq 1$
Determine when $x_n$ is increasing or decreasing , then prove the convergence of $x_n$
if $a > \frac{1}{3} \ then \ x_n \ is \ decreasing$
if $a < \frac{1}{3} \ then \ x_n \ is \ increasing$