lef $f_n\in L^1(\mu)$ be a sequence of non-negative functions converging pointwise to $f\in L^1(\mu)$. prove that:
$\lim_{n \to \infty} (\int_X f_n d\mu - \int_X fd\mu - \int_X|f-f_n|d\mu)=0$
I've tried to use Fatou's lemma on $|f-f_n|-(f-f_n)\ge 0$ and I got to:
$\limsup(\int_X f_n d\mu - \int_X fd\mu - \int_X|f-f_n|d\mu)\le-\int_X\liminf|f-f_n|d\mu$
How do I continue from here?