If $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [-\pi/2,\pi/2]$, find $\lim_{n \to \infty}a_n$ if it exists Let $a_{n+1}=\cos(a_n)$ for $n\ge0$ and $a_0 \in [0,\pi/2]$
Find $\lim_{n \to \infty}a_n$ if it exists.
I drew some sketches and it does seem like the limit exists, it's probably $x$ such that $\cos(x)=x$
I have no idea how to go about solving this, hints would really be appreciated.
Thank you for your time!

The "Fixed Point" method in numerical analysis says that if a function is continuously differentiable in a neighbourhood of a fixed point $x_0$ and 
$$
|f'(x)| \lt 1
$$
it will converge to a fixed point
source: Fixed point
