how to prove this limit, convolution?

Question

I am wondering how to prove that $\vert (f*g)(x) \vert \rightarrow 0 $ when $\vert x \vert \rightarrow \infty$ if we assume $f \in L^{p}(\mathbb R)$ and $g \in L^{q}(\mathbb R)$ where $1/p+1/q=1$ with $p>1$. I think that the idea is to use the density of compactly smooth functions in $L^p$ but I have some trouble to find the solution... Thanks.