Squeeze Theorem for Absolutely Convergent Series

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

Suppose that:


 * $\displaystyle -\sum \size {a_n} = \sum \size {a_n}$

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

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


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

implies that of:


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

By Negative of Absolute Value:


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

By repeated application of Triangle Inequality:


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

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

Hence the result, from Squeeze Theorem for Real Sequences.