Almost complete proof that $\int_A f_n \to \int_A f$

\begin{align*} \int f\chi_{A}\leq\liminf\int f_{n}\chi_{A}\leq\limsup\int f_{n}\chi_{A}. \end{align*} While \begin{align*} \limsup\int f_{n}\chi_{A}&=\limsup\left(\int f_{n}-\int f_{n}\chi_{A^{c}}\right)\\ &\leq\limsup\int f_{n}+\limsup\left(-\int f_{n}\chi_{A^{c}}\right)\\ &=\lim\int f_{n}-\liminf\int f_{n}\chi_{A^{c}}\\ &=\int f-\liminf\int f_{n}\chi_{A^{c}}\\ &\leq\int f-\int f\chi_{A^{c}}\\ &=\int f\chi_{A}. \end{align*}

$|\int_A f_n -\int_A f| \leq \int_X |f_n-f|$. Let us show that $\int_X |f_n-f| \to 0$. $(f-f_n)^{+}\to 0$ almost everywhere and $0 \leq (f-f_n)^{+} \leq f$. By DCT we get $\int_X (f-f_n)^{+}\to 0$. Now $\int_X (f-f_n)^{-}=\int_X (f-f_n)^{+} -\int_X (f-f_n) \to 0-0=0$. Hence $\int_X |f-f_n|=\int_X (f-f_n)^{+}+\int_X (f-f_n)^{-} \to 0$.