proof that $|g|$ divides $\exp G$

The following is a proof that, for every group $G$ that has an exponent and for every $g\in G$, $|g|$ divides $\exp G$.

Proof.

By the division algorithm, there exist $q,r\in \mathbb{Z}$ with such that $\exp G=q|g|+r$. Since $e_G=g^{\exp G}=g^{q|g|+r}={(g^{|g|})}^q{g}^r={(e_G)}^q{g}^r=e_G{g}^r=g^r$, by definition of the order of an element, $r$ cannot be positive. Thus, $r=0$. It follows that $|g|$ divides $\exp G$.∎