Let $f_1, f_2, \ldots$ be a sequence of continuous positive functions of $[0,1]$ and let $a_n = \sup\{ f_n(x) : x \in [0,1]\}$.
In class, we showed that if $\sum f_n$ uniformly converges on $[0,1]$, then it is not always true that $\sum a_n < \infty$.
We did this by visually constructing a sequence of continuous functions of disjoint support on $[0,1]$ such that $a_n = \frac{1}{n}$.
My question is: Does the same conclusion hold if $f_n$ is a decreasing sequence of continuous positive functions on $[0,1]$, or not?