Squeeze Theorem for Absolutely Convergent Series

Theorem
Let $\displaystyle \sum \left\vert{a_n}\right\vert$ be an absolutely convergent series in $\R$.

Suppose that:


 * $\displaystyle -\sum \left\vert{a_n}\right\vert = \sum \left\vert{a_n}\right\vert$

Then $\displaystyle \sum a_n$ equals the above two series.

Proof
From Absolutely Convergent Series is Convergent, the convergence of:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \left\vert{a_n}\right\vert$

implies that of:


 * $\displaystyle \sum_{n \mathop = 1}^\infty a_n$

By Negative of Absolute Value:


 * $\displaystyle -\left\vert{\sum_{n \mathop = 1}^j {a_n} }\right\vert \le \sum_{n \mathop = 1}^j {a_n} \le \left\vert{\sum_{n \mathop = 1}^j {a_n} }\right\vert$

By repeated application of Triangle Inequality:


 * $\displaystyle -\sum_{n \mathop = 1}^j \left\vert{a_n}\right\vert \le \sum_{n \mathop = 1}^j {a_n} \le \sum_{n \mathop = 1}^j \left\vert{a_n}\right\vert$

By hypothesis, the leftmost and rightmost terms converge as $j \to +\infty$.

Hence the result, from Squeeze Theorem for Sequences of Real Numbers.