Prove that in any group, an element and its inverse has the same order.

My Proof:

Let G be a group, and suppose that $\displaystyle a \in G$ and $\displaystyle a^{-1} \in G$.

By definitions, $\displaystyle <a> = n $ such that $\displaystyle a^{n} = e$ and $\displaystyle <a^{-1} = m $ such that $\displaystyle (a^{-1})^{m} = e$

thus, $\displaystyle a^{n} = a^{-m}$

Now, can I say that n must equal to m because the negative of inverse equals to the original power?