Let $(\Omega, \mathcal{A}, \mu)$ be a measurable space with $\mu(\Omega) < \infty$. Now, let $f_n$ be a sequence of functions such that $f_n \rightarrow 0$ in $L^\infty(\mu)$. Does this imply that $f_n \rightarrow 0$ in $L^p(\mu)$ for any $p < \infty$?
If so, what is a formal argument, or even better, a reference for this statement?