Cauchy Product of Absolutely Convergent Series

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Theorem

Let $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ be two real series that are absolutely convergent.


Then the Cauchy product of $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ is absolutely convergent.


Proof


Sources