Equivalence of Sobolev Norms via Fourier Transformation

Question

I am working through a proof which shows that the usual Sobolev norm $\|\cdot\|_{k,2}$ on $W^{k,2}(\Omega)$ is equivalent to the norm

$$ \|u\|_{H^m} := \| (1 + |\cdot|^2)^{\frac{k}{2}}\mathcal{F}u \|_2 $$

At one point we use the inequality

$$ 2^{-k} \leq \frac{1 + |\xi|^{2k}}{(1 + |\xi|^2)^k} \leq 2. $$

Since the fraction always is smaller than one, the right inequality is not a problem. But I can't find a way to prove the left inequality. Is there an elementary proof to this inequality?

Answer 1

Yes. Distinguish the cases $|\xi|\leq 1$ and $|\xi|> 1$ and use crude bounds.