Is $x_{n+1}=\cos x_n$ convergent?

If $x\in \Bbb R$ ; then is the sequence $\{a_n\}$ where $a_1=x$; $a_{n+1}=\cos (a_n)$ convergent?

Obviously $|a_n|\le1$ and hence $(a_n)$ is bounded.

Also $f(x)=\cos x$ is decreasing for $x>0$.

But here $x\in \Bbb R$ .How to proceed here?Please help.

Solution 1:

Note that since $|a_n| \le 1$ we have $a_n \in [0,1]$ since $\cos$ is positive on $[-1,1]$.

Since $\cos$ is decreasing on $[0,1]$ we see that $a_n \ge \cos 1 >0$ for all $n \ge 2$.

In particular, $a_n \in [\cos 1, 1]$ for $n \ge 2$ and we have $|\cos'x| \le \sin 1 < 1$ for $x \in [\cos 1, 1]$ and so $\cos$ is a contraction map on $[\cos 1, 1]$. Hence $a_n$ converges to the unique fixed point.