a. If $\sum_{n\ge 1}g_n$ converges uniformly then $(g_n)$ converges uniformly to zero
Im not sure because if $|\sum_{n=m}^{m+k}g_n|<\varepsilon$ comes from an alternating series then $|g_m-g_{m+k}|<\varepsilon$ dont necessarily holds for the same $\varepsilon$. What Im really sure is that $(g_n)$ must converge to zero to assure the possibility that the series converges.
Can you help me to clarify this question? If Im right, we know some counterexample?
b. If $\sum_{n\ge 1}f_n$ converges uniformly on $A$, then there exists constants $M_n$ such that $|f_n(x)|\le M_n$ and $\sum_{n\ge 1}M_n$ converges.
I think is false, but Im again unsure. Im unable to find some $M_n$ for the uniformly convergent $\sum_{n\ge 1} x^n,\, x\in(-1,1)$. The problem that I see here is that the interval $(-1,1)$ is open and for $|x|=1$ the series diverges, so I cant define any $M_n$ because the supremum for this series is $1$.
Can you help me here please? Thank you in advance.