I'd like your help with proving or refuting the following claim:
If for every $n \in \mathbb{N}$, $f_n:[a,b] \to \mathbb{R} $ is an increasing function, and if $f_n \to f$ pointwise in $[a,b]$ and $\sum \limits_{n=1}^{\infty}f_n(a)$ and $\sum \limits_{n=1}^{\infty}f_n(b)$ converge, then $f_n\to f$ uniformly in $[a,b]$.
I didn't come to any smart conclusion.