Powers of Permutation Element

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

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

Let $$k \in \mathbb{Z}: k > 0$$ 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 $$r = 0$$.