How to prove $f_n \in L^1$

Question

I was trying to build a scheme to solve this kind of question:

Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$.

First of all I need to check that both $f_n$ anf $f:=\lim_{n\to \infty}f_n$ $\in L^1(D)$ and in that case the only possible limit for $f_n$ in $L^1(D)$ is $f$.

After that step I think to know just this 3 ways:

1. use the definition and solve (or compare) the integral $\int_D \left|f_n(x)-f(x)\right|dx$: in this case the $\lim_{n\to \infty} \int_D \left|f_n(x)-f(x)\right|dx$ should be $<+\infty$ or $0$?;

2. use the monotone convergence theorem and check that $\lim_{n\to \infty} \int_D \left|f_n(x)\right|dx < +\infty$;

3. if the dominated convergence theorem can be applied to the $f_n$ I have the convergence.

Is that right? Have you any other hint?

Thanks!