(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.∎

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