Let $\{f_n \}$ be a sequence of upper-semicontinuous functions from some open subset $U \subset \mathbb{C}$ into $\mathbb{R}$ for which $\{ f_n \}$ is uniformly bounded on compact subsets of $U$. Let $\{c_n\}$ be a sequence of positive real numbers such that $\sum_{n=1}^{\infty} c_n < \infty$. Show that $f := \sum_{n=1}^{\infty} c_n f_n$ is upper-semicontinuous.
I can get that finite linear combinations of upper-semicontinuous functions is again upper-semicontinuous (provided we only use positive constants) and that the sum converges absolutely and uniformly on compact subsets of $U$. What I can't seem to do is extrapolate from the finite case to the infinite case.
Best Answer
Let $(s_n)$ be the partial sum sequence of the series. By M-test the series converges uniformly, so $s_n \to f$ uniformly. Let us show that $U=\{x: f(x) <a\}$ is open for any real number $a$. Pick any point $x \in U$. Let $f(x)<b<a$. Then there exists an integer $m$ such that $|s_m(y)-f(y)| <\frac {a-b} 3$ for all $y$. In particular $s_m(x) <f(x)+\frac {a-b} 3<b+\frac {a-b} 3=\frac {a+2b} 3$. Since $s_m$ is u.s.c. there exist an open interval $I$ around $x$ such that $s_m(y) <\frac {a+2b} 3$ for all $y \in I$. Now note that $f(y) <s_m(y)+\frac {a-b} 3<\frac {a+2b} 3+\frac {a-b} 3<a$. Hence $y \in U$ whenever $y \in I$. We have proved that every point of $U$ is an interior point, so $U$ is open.