I needed to find a counter example to "The infinite sum of upper semicontinuous functions is upper semicountinuous". I did find one online, but I wanted to know whether or not mine works.
Let $\{f_n\}$ be indicator functions of closed sets $E_n=[n-0.1, n+0.1]$. Consider the set $U=\{x \mid \sum f_n < 1.5\}$. The point $x=1.1$ is in $U$, since $\sum f_n(1.1)=f_1(1.1)=1<1.5$, so $E_1$ is in $U$. I cannot find an open ball, centred at $x=1.1\in E_1$, which is contained in $U$, so $U$ is not open and $\sum f_n$ is not upper semicontinuous.
Your counter example does not work because your $\sum f_n$ is uppersemicontinuous, as the indicator function of a closed set.
Instead, you can take $f_0=-|\sin|$ and $f_n=|\sin|^n-|\sin|^{n+1}$ for $n\in\mathbb N^*$.