Does this function series converge uniformly?

Question

Prove/Disprove: If the functions $f_n (x)$ are monotonic for every $x \in [a,b]$ and $\sum_{n=1}^\infty f_n (a)$ and $\sum_{n=1}^\infty f_n (b)$ absolutely convergent, then $\sum_{n=1}^\infty f_n (x)$ is absolutely and uniformly convergent on [a,b].

Answer 1

It's true, because you have normal convergence.

Proof: Let's note $f_n(a)=m_n$, $f_n(b)=M_n$ for each $n$. Since $f_n$ is monotonic for each $n$, you can say that $$ \vert\vert f_n \vert\vert_{\infty} = \max(\vert(m_n\vert,\vert M_n\vert) \leq \vert m_n\vert + \vert M_n\vert $$

and the hypothesis is that $\sum_n \vert m_n\vert$ and $\sum_n \vert M_n\vert$ are absolutely convergent. So the inequality above shows that $\sum_n \vert\vert f_n \vert\vert_{\infty}$ converges, i.e. $\sum_n f_n $ converges normally, therefore uniformly on $[a,b]$.