Definition. $x \in \mathbb{R}^n$ is called a fixpoint of the function $f:\mathbb{R}^n \to \mathbb{R}^n$, if $f(x)=x$.
Notation. $f^n(x) := \underbrace{(f \circ f \circ \cdots \circ f)}_{\text{$n$ times}}(x)$
Definition. A function $f:\mathbb{R}^n \to \mathbb{R}^m$ is called a contraction, if there exists a positive constant $L<1$ such that for all $x,y \in \mathbb{R}^n$ the inequality $$\|f(x)-f(y)\|_{\mathbb{R}^m} \leq L \cdot \|x-y\|_{\mathbb{R}^n} $$ holds.
Let $A \subseteq \mathbb{R}$ be a closed subset and $f : A \to A$. Assume, there exists an $n \in \mathbb{N}$, such that the $n$-fold composition $f^n:A \to A$ is a contraction. Can it be shown, that there exists a fixpoint $x^* \in A$ for $f$?