I'd like to refer the following answer:
https://math.stackexchange.com/a/628992/130682
@robjohn claims that:
$$\cos(x):\left[\frac1{\sqrt2},\frac\pi4\right]\mapsto\left[\frac1{\sqrt2},\frac\pi4\right]$$
$\pi\over 4$ is $a_1$ but where does $1\over \sqrt(2)$ came from?
Update:
My actual question is:
Given $a_1 = {\pi \over 4}$, $a_n = \cos(a_{n-1})$
Why does the range of this recurrence is $\left[\frac1{\sqrt2},\frac\pi4\right]$
Thanks.