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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?

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

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    I think as $\int_{E}f_{n} \leq \int_{E}f \implies liminf\int_{E}f_{n} \leq \int_{E}f $ ?? – 2017-02-22
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    Actually it's the other way around: $liminf \le limsup$. – 2017-02-22
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    will edit , ok then?? – 2017-02-22
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    ok,then $liminf\int_{E}f_{n} \leq limsup\int_{E}f_{n} \leq \int_{E}f $ and hence follows right, i made mistake in the inequality , ok, thanks! – 2017-02-22

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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 $\int f.$

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    I think the inequality is incorrect regarding liminf and limsup? – 2017-02-22
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    The point is that since he showed $\limsup\le \liminf$, this means that the $\lim $ itself exists and is equal to $\int f$. – 2017-02-22
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    First of all $\liminf \leq \limsup $!! – 2017-02-22
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    Yes and thus the fact that Royden shows that $\limsup\le \liminf$ means equality holds. – 2017-02-22