Prove the following generalization of Fatou's Lemma: if $\{f_n\}$ is a sequence of nonnegative measurable functions on $E$, then $\int_E \lim \inf f_n \le \lim \inf \int_E f_n$.
I found this, but I don't understand how or why the Monotone Convergence Theorem is of any help. Here's my proof with just relies on Fatou's Lemma.
By definition, $\inf_{k \ge n} f_k$ converges pointwise to $\lim \inf f_n$, so by Fatou's lemma we get
$$\int_E \lim \inf f_n \le \lim \inf \int_E \inf_{k \ge n} f_k$$ But $\inf_{k \ge n} f_k \le f_n$ for every $n \in \Bbb{N}$ which implies $\int_E \inf_{k \ge n} f_k \le \int_E f_n$ for every $n \in \Bbb{N}$ and finally
$$\lim \inf \int_E \inf_{k \ge n} f_k \le \lim \inf \int_E f_n$$ The above two inequalities yield
$$\int_E \lim \inf f_n \le \lim \inf \int_E f_n,$$ which concludes the proof.
Have I made a mistake that is presently eluding me?