TITLE: Convergence of $x_n = \cos (x_{n-1})$
QUESTION [5 upvotes]: I define the sequence $x_n = \cos (x_{n-1}), \forall n > 0$.
For which starting value of $x_0 \in \mathbb{R}$ does the sequence converge?

REPLY [2 votes]: This is just Banach's fixed point theorem, and it is appliable since
$$\forall\,x_1\,,\,x_2\in\Bbb R\,\,\,,\,\,\,|\cos x_1-\cos x_2|\stackrel{\text{M.V.T. for}\,\cos x}=|\sin c|\,|x_1-x_2|\leq |x_1-x_2|$$
for some $\,c\in\,(x_1\,,\,x_2)\,$