Let $( X, \mathcal{M}, \mu)$ be a measure space, let $\{f_{n}\}$ be a measurable sequence of functions, and let $f$ also be measurable. In addition suppose $\mu$ is $\sigma$-finite and that: $$\int_{X} |f_{n} - f|d\mu \to 0 \text{ as } n \to \infty$$
Does it follow that $f_{n}$ and $f$ are integrable functions? i.e. that $f_{n} \in L^{1} (\mu)$, $\forall n$, and $f \in L^{1} (\mu)$?
I need this for a proof I am doing, but I cant for the life of me find a way to prove it.