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.


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