I'm having trouble showing that if $\{f_n\}$ is a sequence of lower semicontinuous functions defined on $\mathbb{R}$ and are non-negative, then $\sum_{n=1}^\infty f_n$ is also lower semicontinuous.
I have a counterexample that works to show why this does not hold for upper semicontinuous functions and I can see why I can't modify it to work for lower semicontinuous ones. I also have shown that the above statement holds for exactly two functions, but can't see how to generalize it. (Another source I consulted simply said that it "trivially follows" from the two-function case, which I found rather unhelpful.)
Any insight/help is much appreciated! Thanks in advance!
(This is a part of Rudin's Real and Complex Analysis Chapter 2 Problem 2.)
Best Answer
Let $u \in \mathbb{R}_+$. We want to show that $U = \left\{ x : \sum_{n \geq 1} f_n(x) > u \right\}$ is open.
If $x \in U$, then $\sum_{n \geq 1} f_n(x) = \sup_{N \geq 1} \sum_{n=1}^N f_n(x) > u$, so there is some $N_x \geq 1$ such that $\sum_{n=1}^{N_x} f_n(x) > u$. Hence $x \in U_{N_x} \subset U$, where the subset $U_{N_x} := \left\{ y : \sum_{n = 1}^{N_x} f_n(y) > u \right\}$ is open (a finite sum of lower semicontinuous maps being lower semicontinuous).