Difference of Absolutely Convergent Series

Theorem
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty \left({a_n - b_n}\right)$ is absolutely convergent, and:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \left({a_n - b_n}\right) = \sum_{n \mathop = 1}^\infty a_n - \sum_{n \mathop = 1}^\infty b_n$

Proof
The series $\displaystyle \sum_{n \mathop = 1}^\infty \left({-b_n}\right)$ is absolutely convergent, as $\left\vert{-b_n}\right\vert = \left\vert{b_n}\right\vert$ for all $n \in \N$.

Then: