Question regarding specific steps of Fatou's lemma proof. I have a question regarding to the following two steps of the proof.

*

*How can we get $\int g_n \leq \int f_n$, or  $g_n \leq f_n$ for all n,  Since by definition of liminf, $g_n \leq f_m $ for all $m \geq n$. Where did the $m$ go?


*Why did $\int f_n$ become $\liminf\int f_n$  suddenly?

 A: For your first question, since $g_n \leq f_m$ for all $m \geq n$, then, if $m = n\implies m=n \geq n$, so $g_n \leq f_n$.
The second question, what happened was that he passed the liminf in the inequality:
$$\int g_nd\mu \leq \int f_n d\mu \implies
lim\inf_n \int g_nd\mu \leq lim\inf_n\int f_n d\mu  $$
Now, since $g_n$ is increasing, then
$$lim\inf_n \int g_nd\mu =lim_n \int g_nd\mu $$ Therefore, you get
$$lim_n \int g_nd\mu \leq lim\inf_n\int f_n d\mu $$
A: Just write the set $\{f_i : n\leq i < \infty\}$ explicitly, i.e. $\{f_n,f_{n+1},...\} \forall n\geq 1$.
Then you take infimum over this set, i.e $g_n$
So, it is obvious that $g_n\leq f_n\ \forall n\geq 1$ , then take the integral both side, we get
$$
\int g_n d\mu\leq \int f_n d\mu.............(*)
$$
Now concentrate on $g_n$.
$g_n \leq g_{n+1}$ and $\lim g_n = \liminf\{f_i : n\leq i < \infty\}= \liminf f_n = f$, this is from definition of limit infimum of sequence.
Then use Monotone convergence theorem on ${g_n}$.
We get $$\lim \int g_n d\mu = \int f d\mu \\
 \Rightarrow \liminf\int g_n d\mu = \int f d\mu
$$
since $\lim \int g_n$ exists and $\lim \int g_n = \liminf\int g_n d\mu$.
now take $\liminf$ both side of (*) you get the desired result.
