Order of Group Element not less than Order of Power/Proof 2

Proof
Let $g^n = e$.

Let $h = g^m$.

Then:
 * $h^n = g^{mn} = \left({g^n}\right)^m = e^m = e$

Hence by definition of order of group element:
 * $\left\lvert{h}\right\rvert \le n$

Hence the result.