## Fall 2017, problem 51

Let $f$ be a bijective function from $A = \left\{1,2, \cdots, n \right\}$ to itself. Show that there is a positive integer $M$ such that $f^M(i) = f(i)$ for each $i$ in $A$, where $f^M$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.