Powers of Permutation Element

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\pi \in S_n$, and let $i \in \N^*_n$.

Let $k \in \Z: k > 0$ be the smallest such that:
 * $\pi^k \left({i}\right) \in \left\{{i, \pi \left({i}\right), \pi^2 \left({i}\right), \ldots, \pi^{k-1} \left({i}\right)}\right\}$

Then $\pi^k \left({i}\right) = i$.

Proof
Suppose $\pi^k \left({i}\right) = \pi^r \left({i}\right)$ for some $r > 0$.

Then, since $\pi$ has an inverse, $\pi^{k - r} \left({i}\right) = i$.

This contradicts the definition of $k$, so $r = 0$.