This is Theorem 8.54, the Sobolev Embedding Theorem, in Folland's Analysis: If $s>k + (n/d)$, then $H_s \subset C_0^k$. (Here $H_s$ is defined to be the Sobolev space and $C_0^k = \big\{f \in C^k(\mathbb{R}^n:\partial^\alpha f \in C_0 \text{ for } |\alpha | \leq k\big\}$).
Proof: Suppose $f \in H_s$; by the Fourier inversion theorem and the Riemann-Lebesgue lemma, it will suffice to show that $\widehat{\partial^\alpha f} \in L^1$ for $|\alpha| \leq k$.
He then goes on to prove this. However, why does it suffice to show that $\widehat{\partial^\alpha f} \in L^1$?
Reference: I am using the following copy of Folland found online: here