Proof

Let $\sum_{k=0}^\infty x_k$ and $\sum_{k=0}^\infty y_k$ be real infinite series.
By hypothesis, let $\sum_{k=0}^\infty y_k$ be convergent.
By the Cauchy criterion, for every $\epsilon > 0$ there is an index $N(\epsilon)\in\mathbb N$ such that \[\left| \sum_{k=m}^n y_k \right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
Since, by hypothesis, $|x_k|\le y_k$ for all $k\in\mathbb N,$ we have $$\left|\sum_{k=m}^n |x_k|\right| \le \left|\sum_{k=m}^n y_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).$$
Applying the Cauchy criterion once again, we get that the $\sum_{k=0}^\infty |x_k|$ is convergent.
By definition, $\sum_{k=0}^\infty x_k$ is absolutely convergent.
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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983