The following question is taken from Royden Real Analysis $4$th edition, Chapter $4,$ question $20.$
Let $\{f_n\}$ be a sequence of nonnegative measurable functions that converges to $f$ pointwise on $E.$ Let $M\geq 0$ be such that $\int_E f_n\leq M$ for all $n.$ Show that $\int_E f\leq M.$ Verify that this property is equivalent to the statement of Fatou's Lemma.
I have proven that $\int_E f\leq M.$ But I have no idea on how to tackle second part.
After some googling, I found a solution in MSE, which goes as follows:
Let $(f_n,n\in\Bbb N)$ be a sequence of measurable integrable functions and $a_N:=\inf_{k\geqslant N}\int f_kd\mu$. Working with the sequence $(f_n,n\geqslant N)$ (for which the sequence of integrals has the same $\liminf$ as those of the whole sequence), one can see that that $\int fd\mu\leqslant a_N$ for each $N$. Now take the limit $\lim_{N\to +\infty}$.
Questions:
$(1)$ What is a motivation of considering $a_N:=\inf_{k\geq N}\int f_k d\mu?$
$(2)$ Why does $\int f d\mu\leq a_N$ for each $N?$
$\displaystyle\int\inf_{k\geq N}f_{k}d\mu\leq\int f_{k}d\mu$ for all $k\geq N$, so $\displaystyle\int\inf_{k\geq N}f_{k}d\mu\leq a_{N}\leq M:=\sup_{N}a_{N}$ for all $N=1,2,...$. Now $\inf_{k\geq N}f_{k}\rightarrow\liminf_{n}f_{n}$ as $N\rightarrow\infty$, by the result, then $\displaystyle\int\liminf_{n}f_{n}d\mu\leq M$, and it is actually the case that $M=\sup_{N}a_{N}=\sup_{N}\inf_{k\geq N}\displaystyle\int f_{k}d\mu=\liminf_{n}\displaystyle\int f_{n}d\mu$.