I am self studying Apostol Mathematical Analysis Chapter->Lebesgue Integration and I was unable to think about an argument used in that proof.
Adding it's image ->
Can someone please tell a rigorous argument which deduces the blue underlined portion of the proof.
$G_{n,1}(x)=\max \{f_1(x),f_2(x),...,f_n(x)\}$. The right hand side is increasing and its limit is $\sup \{f_n(x): n=1,2...\}$. Also $G_{n,1}(x) \to G_1(x)$ almost everywhere. Hence you only have to take limit as $n \to \infty$ to get $G_1(x)=\sup \{f_1(x),f_2(x),...\}$ almost everywhere.