(related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
Let \(\sum_{k=0}^\infty x_k\) be an absolutely convergent series. Since, be definition, \(\sum_{k=0}^\infty |x_k|\) is convergent, we can apply Cauchy's general criterion for convergent series and conclude that for every \(\epsilon > 0\) there is an index \(N(\epsilon)\in\mathbb N\) such that
\[\left|\sum_{k=m}^n |x_k|\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
According to the triangle inequality, we have that
\[\left|\sum_{k=m}^n x_k\right| \le \left|\sum_{k=m}^n |x_k|\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
Applying Cauchy's general criterion for convergent series once again, we get that the series \(\sum_{k=0}^\infty x_k\) is convergent (in the usual sense).
(to be done: got the proof - become a co-author?)