if $|f_n|<g \in L^1$, and $f_n \rightarrow f$ in measure, how do we know $\lim_{n\to \infty} \int f_n = \int f$

Question

I know that a subsequence converges, but I am not even convinced that $\int f_n$ converges at all. They are all finite, but I am not certain how to bound them. I have considered working with $\int (f-f_n)$ and things like that but I still don't see that we have anything that would let me apply the Dominated Convergence Theorem except to a subsequence.

Answer 1

Use the following fact:

If every subsequence has a subsequence that converges to $a$, then the sequence converges to $a$.