Suppose that $(f_n)$ is a sequence of measurable functions on $E \subseteq \mathbb{R}$ such that
$$\sup_n \| f_n \|_p < \infty\ \mbox{ for some } p > 1,$$
where
$$\|g\|_p := \Big(\int_{\mathbb{R}}|g(x)|^p dx \Big)^{1/p}.$$
Show that
$$\lim_{N \rightarrow \infty} \sup_{n \geq1} \int_{\{x \in E : |f_n(x)|>N\}}|f_n(x)|dx = 0.$$
Attempt. I am able to show that for all $n$,
$$\lim_{N \rightarrow \infty} \int_{\{x \in E : |f_n(x)|>N\}}|f_n(x)|dx = 0.$$
But what I need is uniform convergence and I am not sure how to achieve that.