While self studying Lebesgue integration from Tom M Apostol I am unable to think about an how to deduce a line in the proof whose image I am adding( line is highlited.
I am not able to deduce the line that the sequence { $G_n(x) $ } decreases almost everywhere and hence converges everywhere.
I am unable to get how the author wrote that {$ G_n (x) $ } is decreasing? and how that implies that it's convergent.
Kindly help.
$\{G_n (x)\}$ is decreasing because it's defined as supremum of $\{f_n(x), f_{n+1}(x),...\}$. $G_1 (x)=\sup\{f_1(x),f_2(x),f_3(x),...\}$ and $G_2(x)=\sup\{f_2(x), f_3(x),...\}$, so, depending on $f_1(x)$, $G_2(x)\leq G_1(x)$ and so on. And this is why $\{G_n (x)\}$ converges a.e.: $\{f_n(x)\}$ converges a.e. and for fixed $x_0$, such that number sequence $\{f_n(x_0)\}$ converges, we have $$\lim_{n\to \infty}f_n(x_0) = \lim_{n\to \infty}\left( \sup_{m\geq n}f_n(x_0)\right) = \lim_{n\to \infty}\left( \inf_{m\geq n}f_n(x_0)\right)$$ So, $$\lim_{n\to \infty}f_n(x_0) = \lim_{n\to \infty}G_n(x_0)$$ hence $\{G_n (x)\}$ converges in points, where $\{f_n (x)\}$ also converges.