Suppose that $f_n$ is a sequence of real measurable functions on a set $X$ of finite measure, and suppose that there is some $\epsilon$ such that for all $n\geq1$:
I want to show that this is false: $f_n(x)\rightarrow f(x)$ a.e. on $X$.
I tried to prove the existence of a set $Y\subseteq X$ of positive measure such that $f_n(x)$ is far from $f(x)$ for all $x\in Y$. However, I only obtain sets $Y_n$ that depend on $n$ so I guess I need some stronger argument. Any help is appreciated!