The following statement is from the book " The analysis of linear partial diffential operators I by Lars Hörmander page 252
If $v \in \mathcal{E}'(\mathbb{R^n})$ ( $v$ is a distribution on $\mathbb{R}^n$) we can decide whether $v$ is in $C_0^\infty $ by examining the behavior of the Fourier transform $\hat{v}$ at $\infty$ . In fact, if $v \in C_0^\infty$ then $$ |\hat{v}(\xi)| \leq C_N {(1+ |\xi|)}^{-N}, \quad N =1,2,..., \xi \in \mathbb{R}^n,$$ By lemma 7.1.3. Conversely, if the inequality above is fulfilled then $v \in C_0^\infty$ by Fourier's inversion formula.
Why does the Fourier's inversion formula imply that $v \in C_0^\infty$? And why we could decide whether $v$ is in $C_0^\infty$ by examining the behavior of the Fourier transform $\hat{v}$ at $\infty$?
Your help would be greatly appreciated!