Powers of Infinite Order Element

Theorem
Let $$G$$ be a group whose identity is $$e$$.

Let $$a \in G$$ have infinite order in $$G$$.

Then $$\forall m, n \in \mathbb{Z}: m \ne n \Longrightarrow a^m \ne a^n$$.

Proof
Let $$m, n \in \mathbb{Z}$$. Then:

$$ $$ $$ $$ $$

The result follows from transposition.