At the moment, I am studying for an exam and I came across the following exercise:
Consider the map $f:[0,1] \to \mathbb{R}$, $f(x)=1-\arctan(x)$. Prove the following statements:
a) $f$ has a unique fixed point
b) For each $x \in [0,1]$, the sequence defined by $$x_0=x,\phantom{aa}x_{n+1}=f(x_n)$$ converges to the fixed point of $f$
I have no problems with a) but I have trouble with b): If I am right, $f$ is not contractive, so we cannot apply the Banach fixed point theorem and that is my problem. (I only know that there is some intervall $[a,b] \subseteq [0,1]$ such that $[a,b]$ contains the fixed point of $f$ and such that the restriction $f|_{[a,b]}$ satisfies the conditions of the Banach theorem but I am not sure whether this might help.) So, how to solve part b)?