Squeeze Theorem for Absolutely Convergent Series

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

Suppose that:


 * $-\sum \left|{a_n}\right| = \sum \left|{a_n}\right|$

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

Proof
From Absolutely Convergent Series is Convergent


 * $\displaystyle \sum_{n=1}^\infty \left|{a_n}\right| \ \text{converges} \ \implies \displaystyle \sum_{n=1}^\infty a_n \ \text{converges}$

By Negative of Absolute Value:


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

By repeated application of Triangle Inequality:


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

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

Hence the result, from Squeeze Theorem for Sequences.