Why if $\phi \in L^1(\mathbb{R}^d)$ then $\displaystyle\int_{\|\mathbf{x}\|>R} |\phi|\mathrm{d}\mathbf{x} < \epsilon $?
I started the proof by:
Suppose that $\exists \epsilon_0 > 0$ and $j_0 \in \mathbb{N}$ such that $\forall j \geq j_0$,
$$\int_{\|\mathbf{x}\|>R_j} |\phi| \mathrm{d}\mathbf{x} \geq \epsilon_0 $$
I don´t know how to continue. I consider using if $\phi(\mathbf{x}) \in L^1(\mathbb{R}^d)$ then $\displaystyle\lim_{\|\mathbf{x}\| \to \infty} |\phi(\mathbf{x})| = 0$.