Suppose for all $\displaystyle n \in \mathbb{N}$, we have $\displaystyle f_n \in L^p(X) $ where $\displaystyle m(X) < \infty$.
Prove that for every $\displaystyle \epsilon>0$ , there exists a set $\displaystyle E$ of finite measure, such that for all $\displaystyle n \in \mathbb{N}$ ,$\displaystyle \int_{E^c} |f_n|^p \, dm < \epsilon$.