Let $\,f, f_n $ be Lebesgue integrable functions mapping reals to extended reals such that, almost everywhere, $\,f_n \to f $. Show that $$\int\left\lvert\,f_n\right\rvert\,d\lambda\to\int\left\lvert \,f\right\rvert\,d\lambda\implies\int\left\lvert\,f_n - f\right\rvert\,d\lambda\to0$$
I have shown the converse by breaking absolute values down into the sum of positive and negative parts. I did not use the almost everywhere convergence condition, so I am expecting that to be important here. It makes me think of the DCT, but I cannot figure out how to use that since there is no (to me) obvious bound.
