Suppose that $\displaystyle f_n\to f$ in measure and $\displaystyle |f_n|\le g\in L^1$, for all $\displaystyle n$. Show that $\displaystyle f_n\to f$ in $\displaystyle L^1$. That is $\displaystyle \lim_n\int_X|f_n-f|d\mu=0$.

I have already shown that $\displaystyle \int_Xfd\mu=\lim_n\int_X f_nd\mu$, but don't see how to use this or anything else to get to the desired result.