$f_n\rightarrow f$ in every $L^p(\mathbb R^d)$

Question

Is this correct? If $f\in L^1(\mathbb R^d)\cap L^\infty(\mathbb R^d)$ then there exists $(f_n)_{n\in\mathbb N}$ in $C^\infty(\mathbb R^d)\cap L^1(\mathbb R^d)\cap L^\infty(\mathbb R^d)$ such that $f_n\rightarrow f$ in $L^p(\mathbb R^d) $ for every $p\in [1,\infty)$.
I don't how to prove or disprove this. Maybe using the theorem of Riesz-Fischer?