(related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)

- By hypothesis, $\sum_{k=0}^\infty x_k$ is a complex series, $\sum_{k=0}^\infty y_k$ is a convergent real series, and $|x_k|\le y_k$ for all $k\in\mathbb N.$
- Since
*\sum_{k=0}^\infty y_k*is a convergent real series, thus we can apply Cauchy's general criterion for the convergence of infinite complex series, and conclude that for every \(\epsilon > 0\) there is an index \(N(\epsilon)\in\mathbb N\) such that $$\left| \sum_{k=m}^n y_k \right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).$$ - Because by hypothesis \(|x_k|\le y_k\) for all \(k\), we get $$\left|\sum_{k=m}^n |x_k|\right| \le \left|\sum_{k=m}^n y_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).$$
- Applying Cauchy's general criterion for the convergence of infinite complex series once again, we get that the \(\sum_{k=0}^\infty |x_k|\) is convergent.
- Therefore \(\sum_{k=0}^\infty x_k\) is an absolutely convergent complex series.∎

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983