Theorem: (Fatou's lemma). Let $(f_n)_{n\geq 1}$ be a sequence of measurable functions with values in $[0,\infty]$. Then $$\int_S\liminf\limits_{n\to\infty}f_nd\mu \leq \liminf\limits_{n\to\infty}\int_S f_n d\mu.$$
Exercise: Assume that $f, f_1, f_2, ...:S\to\overline{\mathbb{R}}$ are measurable functions such that
i) There is a constant $M\geq 0$ such that for all $n\in\mathbb{N}, \int_S\left|f_n\right|d\mu\leq M$.
ii) $f_n\to f$ pointwise.
Use Fatou's lemma to show that $\int_S\left|f\right|d\mu\leq M$.
Solution: Fatou's lemma yields that: $$\int_S\left|f\right|d\mu = \int_S\liminf\limits_{n\to\infty}\left|f_n\right|d\mu\leq \liminf\limits_{n\to\infty}\int_S f_nd\mu \leq M.$$
My question: How do we know that $\int_S\left|f\right|d\mu = \int_S\liminf\limits_{n\to\infty}\left|f_n\right|d\mu$ and $\liminf\limits_{n\to\infty}\int_S f_nd\mu \leq M$?
Thanks in advance!