# Show that if $\lim_{n\rightarrow\infty}\int_E|f_n|=\int_E|f|$ then $\int_E|f-f_n|\rightarrow 0$. [duplicate]

Let $$\{f_n\}$$ be a sequence of integrable functions on $$E$$ for which $$f_n$$ converges to $$f$$ a.e. on $$E$$, and $$f$$ is integrable over $$E$$. Show that if $$\lim_{n\rightarrow\infty}\int_E|f_n|=\int_E|f|$$ then $$\int_E|f-f_n|\rightarrow 0$$.

So I feel like this should be really obvious, but I'm having a hard time proving it. I'd appreciate some advice! The details are what get me!

• not obvious, iirc. go through proof of dominated convergence theorem. – mathworker21 Nov 6 '19 at 0:42
• You may want to check the answer in the link. – Sangchul Lee Nov 6 '19 at 0:53