I need to do the question at the top of this image.
I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim g_n(x) = \liminf f_n(x) = f(x)$.
Hence from MCT, I know $$\int f\ du = \int \lim\ g_n\ du = \lim \int g_n du$$
I can apply Fatou's Lemma since $f_n$ is a sequence of non-negative measurable functions. Hence, I know $$\int\lim inf\ f_n\ du\le \lim\inf \int f_n\ du$$
I need to prove that $$\int f\ du \leq \lim\inf \int f_n\ du$$