# If $\sum g_n(x)$ converges uniformly and absolutely and $|f_n(x)|\leq |g_n(x)|$ show that $\sum f_n(x)$ converges uniformly and absolutely.

I do not know how to prove if the statement above is true. I know i can use the

Cauchy criterion i.e. $$|\sum_{n\rightarrow m}f_n(x)|\leq\sum_{n\rightarrow m}|f_n(x)|\leq \sum_{n\rightarrow m}|g_n(x)|$$.

However, we do not know if the last inequality holds given that only $$\sum g_n(x)$$ converges uniformly and not $$\sum |g_n(x)|$$.

Secondly, what do they mean by converging uniformly and absolutely? Are they referring to $$\sum |g_n(x)|$$ converging uniformly or $$\sum g_n(x)$$ converging uniformly and absolutely point-wise?

This question is with regard to: Question on uniform convergence of sum of continuous functions.

To say that $$\sum f_n$$ is uniformly and absolutely convergent (for $$x \in D$$) without further qualification generally means that there exist functions $$S$$ and $$\hat {S}$$ such that for all $$\epsilon > 0$$ there exists $$N_1 \in \mathbb{N}$$ such that for all $$n > N_1$$ and for all $$x \in D$$ we have

$$\left |\sum_{k=1}^n f_k(x) - S(x) \right| < \epsilon,$$

and there exists $$N_2(x) \in \mathbb{N}$$, which may depend on $$x$$, such that $$n > N_2(x)$$ implies

$$\left |\sum_{k=1}^n |f_k(x)| - \hat{S}(x) \right| < \epsilon$$

However, it is possible that $$\sum f_n$$ converges uniformly, but $$\sum|f_n|$$ converges pointwise but not uniformly.

If $$\sum |f_n|$$ is also uniformly convergent then the correct and unambiguous terminology is that $$\sum f_n$$ is uniformly absolutely convergent.

For the proof in question, "they" are specifying uniform absolute convergence by saying "converging uniformly and absolutely". This facilitates the proof of uniform absolute convergence of $$\sum f_n$$ through the Cauchy criterion using the inequalities

$$\left|\sum_{k=n+1}^m f_k(x) \right| \leqslant \sum_{k=n+1}^m |f_k(x)| \leqslant \sum_{k = n+1}^m |g_k(x)|$$

Otherwise, with complex valued functions $$|f_n(x)| \leqslant |g_n(x)|$$ is not sufficient information to prove uniform convergence of $$\sum f_n$$ if $$\sum|g_n|$$ does not converge uniformly.