Cauchy Product of Absolutely Convergent Series

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.