# A question in Proof of Theorem 10.27 of Apostol Mathematical Analysis

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.

• The fact $G_n(x)$ is decreasing is already explained in "since $A\subseteq B$ implies $\sup A \leq \sup B$". Commented Jun 19, 2020 at 6:29
• The convergence of $G_n(x)$ is explained afterwards by showing that if $f_n(x) \to f(x)$ then $G_n(x) \to f(x)$. The details are in next page of the book. Commented Jun 19, 2020 at 6:33

$$\{G_n (x)\}$$ 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.