Power of Moved Element is Moved

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

Let $\sigma \in S_n$.

Then for all $m \in \Z$:
 * $i \notin \operatorname{Fix} \left({\sigma}\right) \implies \sigma^m \left({i}\right) \notin \operatorname{Fix} \left({\sigma}\right)$.

Proof
Aiming for contradiction, suppose that there exists a $i \notin \operatorname{Fix} \left({\sigma}\right)$ and a $m \in \Z$ such that $\sigma^m \left({i}\right) \in \operatorname{Fix} \left({\sigma}\right)$.

Then:

But it was previously established that $i$ was moved by $\sigma$.

This is a contradiction, therefore $\sigma^m \left({i}\right)$ is moved by $\sigma$.