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Proof

(related to Proposition: Direct Comparison Test For Divergent Series)

• By hypothesis $\sum_{k=0}^\infty x_k$ and $\sum_{k=0}^\infty y_k$ are real infinite series with $x_k\ge y_k\ge 0$ for all $k\in\mathbb N$ and with $\sum_{k=0}^\infty y_k$ being divergent.
• Assume, $\sum_{k=0}^\infty x_k$ is convergent.
  • Then, by the direct comparison test for absolutely convergent series it would follow that $\sum_{k=0}^\infty y_k$ is absolutely convergent.
  • Then, because absolute convergence implies convergence, $\sum_{k=0}^\infty y_k$ would by convergent.
  • This is a contradiction to hypothesis.
• It follows that the assumption is false.
• Therefore, $\sum_{k=0}^\infty x_k$ is not convergent, thus it is divergent.
  ∎

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References

Bibliography

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