Study the sequence defined by the recursion:
$$ u_{0}>-1,\quad u_{n+1}=\sqrt{\frac{1+u_{n}}{2}}. $$
i write
$$ \text{Behavior: }u_{n+1}>u_n\mathrm{~if~}\sqrt{\frac{1+u_n}2}>u_n $$
Monotonicity: We show that if $u_n> - 1, $ then $u_{n+ 1}> u_n$
$$ \text{Solving for monotonicity: }\sqrt{\frac{1+u_n}2}>u_n\Rightarrow\frac{1+u_n}2>u; $$
$$ \Rightarrow1+u_n>2u_n^2\Rightarrow2u_n^2-u_n-1<0 $$
i don't know next step
