Monotone Convergence theorem (Measure theory,Royden Real analysis).

Question

I was reading the proof of monotone convergence theorem, I understood the proof except the ones marked in blue, I know that $\liminf \leq \limsup$, but how they arrive at equality using this, I am unable to understand?

Using $\liminf \leq \limsup$ , how we proceed to prove the equality!

Answer 1

Fatou says

$\tag1 \liminf_n\int f_n\ge \int f.$

On the other hand $\int f_n\le \int f,\ $ so

$\tag2 \limsup_n\int f_n\le \int f.$

Combine $(1)$ and $(2)$ to get

$\tag3 \limsup\int f_n\le \int f\le \liminf_n\int f_n, $

and this implies immediately that $\lim_n\int f_n$ exists and is equal to $\int f.$