Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a contraction, i.e. there exist $0\le\beta<1$ such that $d(S^m(x),S^m(y)) \le \beta d(x,y)$ for all $x,y \in X$.
I want to show that this implies that $S$ has a unique fix-point, i.e. there exist a unique $z \in X$ such that $S(z) = z$.
I've tried to apply Banach fix-point theorem which imply that $S^m$ has a unique fix-point. Also, I've tried consider if $S$ has an inverse. Last, I've tried to come up with a proof by contradiction.
Can anyone guide or show how the above could be proven ?