If $f_n$ is in $L^1$ and $f_n$ converges to $f$, is $f$ in $L^1$?

Question

This question might come off as basic to most of you, but this isn't basic to me.

If $f_n$ is in $L^1$ and $f_n$ converges to $f$ in $L^1$ ($||f_n-f||_1 \to 0$ as $n\to \infty$), does it necessarily imply that $f$ is in $L^1$?