Suppose $\sum_{n=1}^\infty f_{n}(x)$ is a series of monotone functions on $[a,b]$, and if both $\sum_{n=1}^\infty f_{n}(a)$ and $\sum_{n=1}^\infty f_{n}(b)$ converge absolutely, prove $\sum_{n=1}^\infty f_{n}(x)$ converges uniformly on $[a,b]$
If $\sum_{n=1}^\infty f_{n}(x)$ is a series of monotone increasing functions then $|f_{n}(x)|\leq f_{n}(b)$
And since $\sum_{n=1}^\infty f_{n}(b)$ converges absolutely, by Weierstrass M test, $\sum_{n=1}^\infty f_{n}(x)$ converges absolutely and uniformly
If $\sum_{n=1}^\infty f_{n}(x)$ is a series of monotone decreasing functions then $|f_{n}(x)|\leq f_{n}(a)$
And since $\sum_{n=1}^\infty f_{n}(a)$ converges absolutely, by Weierstrass M test, $\sum_{n=1}^\infty f_{n}(x)$ converges absolutely and uniformly
I'm not sure if $M_{n}$ is positive in both cases. Is this prove wrong ?