Problem:
Let $f$ be an $N$ times differentiable complex-valued function on $\mathbb{R}$ and $$\int \left| \; f^{n}(x) \, \right| dx < \infty, \; \; \; \; n = 0, \ldots, N$$ then $\sup\limits_{\xi} \left| \; \hat{f}(\xi)(1+\left| \xi \right|^N)^{-1} \, \right| < \infty$.
My attempt:
$$\sup_\xi \left| \; (1 + |\xi|^N)^{-1} \int_{-\infty}^{\infty} e^{ikx}f(x)dx \, \right| \leq \sup_\xi\left( \left| \; (1+ |\xi|^N)^{-1} \, \right| \cdot \left| \; \int_{-\infty}^{\infty} |f(x)|dx \, \right| \right) < \infty$$
Is this okay?