I'm stuck on how to show this sequence monotonically decreasing: $U_{n+1}=U_{n}^{2}+\frac18$ and $U_0=\frac12$ as infos I have :
$U_{0}=\frac{1}{2}$ 
$U_{n+1}=U_{n}^{2}+\frac{1}{8}$
I have already proved that $U_{n}$ is positive as the exercise requested but I still don't know how to use it with $U_{n+1}$ to show that $U_{n}$ is decreasing monotonically .
Thanks for your attention.

Proceed by induction.

First verify that $u_1 < u_0$.

Next suppose $u_{n}<u_{n-1}$ for some positive integer n. 

As you've already shown, the terms of the sequence are positive, hence
$$u_n<u_{n-1}$$
$$\Rightarrow {u_n}^2 < {u_{n-1}}^2$$
$$\Rightarrow {u_n}^2 + \frac{1}{8} < {u_{n-1}}^2 + \frac{1}{8}$$
$$\Rightarrow u_{n+1} < u_n$$
which completes the induction.
