real analysis - Upper bound of a recursive sequence for a fixed $n$

I have a recursive sequence given by $$x_n = \sqrt{\frac{1+x_{n-1}}{2}},\ x_1=0$$
I can easily show that it is increasing, bounded and thus converges with its limit being $1$. But what if I wanted to upper bound $x_{n_0}$ for some fixed $n_0$? This bound should be dependent on $n$. How should I proceed?

Answer

$$
x_0=\cos \frac{\pi}{2}\\
x_1=\cos \frac{\pi}{4}\\
\vdots\\
x_n=\cos 2^{-n-1}\pi

$$