This question concerns one detailthe given proof of the Riemann-Lebesgue lemma in Folland's Real Analysis - Modern Techniques and Their Applications (2nd edition), pp. 249. The claim is that $\mathcal{F}\left(L^1\left(\mathbb{R}^n\right)\right)\subset C_0\left(\mathbb{R}^n\right)$. The author begins the proof as follows:
By e.), if $f\in C^1\cap C_c$, then $|\xi|\hat{f}(\xi)$ is bounded and hence $\hat{f}\in C_0$.
where the part e.) is
If $f\in C^k$, $\partial^\alpha f\in L^1$ for $|\alpha|\leq k$, and $\partial^\alpha f\in C_0$ for $|\alpha|\leq k - 1$, then $\hat{\left(\partial^\alpha f\right)(\xi)} = \left(2\pi i \xi\right)^\alpha \hat{f}(\xi)$
and $C_c$ is the set of continuous functions with compact support and $C_0$ is the set of continuous functions vanishing at the infinity.
What I can't recall/grasp is that why does the Fourier transform of $f\in C^1\cap C_c$ vanish at infinity?