Let $f \in L^1_{\text{loc}}\left(\mathbb{R}\right)$ be a nonnegative function. Show that if $\Lambda_f$ is a tempered distribution, then there exist $C > 0, N \in \mathbb{N}_0$ satisfying
$$\forall R \geq 1: \int_{-R}^Rf(x)dx \leq C\left(1+R\right)^N$$
My attempt was to pick some $A > 0$ satisfying $|\Lambda_f\left(\phi\right)| \leq |A\nu_N\left(\phi\right)|, \phi\in \mathscr{D}\left(\mathbb{R}\right)$, and fix some $\psi\in \mathscr{D}\left(\left[-2, 2\right]\right)$ such that $\psi = 1$ on $\left[-1, 1\right]$.
So, $\displaystyle0 \leq \int_{-R}^Rf(x)dx\leq\int_{-R}^Rf(x)\psi\left(\frac{x}{R}\right)dx\leq A\nu_N\left(\psi\left(\frac{\cdot}{R}\right)\right) \leq...$ and that's where I got stuck.
Thanks in advance