(related to Proposition: Cauchy-Schwarz Test)

- By hypothesis, the "squared" infinite series $\sum_{n=0}^\infty a_n^2$ and $\sum_{n=0}^\infty b_n^2$ are both convergent, say to the limits $\alpha$ and $\beta.$
- By the Cauchy-Schwarz inequality and the triangle inequality we have for the partial sums the relation $$\left|\sum_{n=0}^N a_n b_n\right|\le \sum_{n=0}^N |a_n b_n|\le \left(\sum_{n=0}^N a_n^2 \right)^{\frac 12}\left(\sum_{n=0}^N b_n^2 \right)^{\frac 12}$$ for all $N\ge 0.$
- Because convergence preserves order relation, we have that the sequence $(s_N)_{N\in\mathbb N}$ of partial sums $s_N:=\sum_{n=0}^N |a_n b_n|$ is bounded from above and by definition, monotonically increasing.
- Since every bounded monotone sequence is convergent, so is $(s_N)_{N\in\mathbb N}$, and thus the series $\sum_{n=0}^\infty |a_n b_n|$ is convergent.
- It follows that $\sum_{n=0}^\infty a_n b_n$ absolutely convergent.∎

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