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Proof

(related to Corollary: Monotony Criterion for Absolute Series)

• By the definition of absolute value, the terms $|x_k|\ge 0$ are all non-negative.
• Therefore, by the monotony criterion, the series $\sum_{k=0}^\infty |x_k|$ is convergent if and only if the sequence $(s_n)_{n\in\mathbb N}$ of its partial sums $s_n:=\sum_{k=0}^n |x_k|$ is bounded.
• By definition of absolutely convergent series, the series $\sum_{k=0}^\infty x_k$, is absolutely convergent if and only if the sequence $(s_n)_{n\in\mathbb N}$ is bounded.
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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition