Prove $n^2 < 2^{n+1}$ by well-ordering

Question

Please help me prove $n^2 ≤ 2^{n+1}$ by the well-ordering (not induction) $n\in\Bbb N \setminus \{{0\}}$

I've assumed that $n=1$ is not counter example so $m > 1$ and now try with $n = m-1$.

$$(m-1)^2 ≤ 2^{m-1+1} = 2^m$$

I'm stuck here.