Let $(f_j)_{j \in \mathbb{N} } \rightarrow f$ all be measurable, and positive functions ($f$ too). Assume $|f_n| \le f$ for all n. Show $\lim_{j \rightarrow \infty} \int f_j \ d \mu = \int f \ d \mu$.
My attempt: since the limit exists, $f$ is also the $\liminf$ and $\limsup$ of the $f_j$s. If we assume for the moment that $f$ is integrable, then the conditions for Reverse Fatou's Lemma are satisfied. If we apply first Fatou, then Reverse Fatou, we get $$\int f \ d \mu \le \liminf \int f_j \ d \mu \le \limsup \int f_j \ d \mu \le \int \limsup f_j \ d \mu = \int f \ d \mu$$ First inequality is Fatou, third is Reverse Fatou. So all inequalities are equalities, and in particular the 2nd and 3rd expressions are equal. Thus the limit exists, and equals the integral of $f$.
If $f$ is not integrable, its integral must be positive infinity. Then we cannot apply Reverse Fatou's lemma, but Fatou's lemma still holds: $$\infty = \int f \ d \mu = \int \liminf_{j \rightarrow \infty} f_j \ d \mu \le \liminf_{j \rightarrow \infty} \int f_j \ d \mu \le \limsup \int f_j \ d \mu$$ So since both the $\liminf$ and $\limsup$ of the integral of the $f_j$s are greater than infinity, they must equal infinity, and so must the limit. This is infinity too, and we are done.
Is above proof correct? If there are mistakes, then I am on the right track, or have I gone in the completely wrong direction?
I think your proof is basically correct but once you've used Fatou to get
$$ \int f \leq \liminf \int f_j$$
you can finish off quickly: for each $j$ we have
$$ \int f_j \leq \int f$$
(since $f_j \leq f$) so the $\limsup$ of the LHS is at most $\int f$.