Product of Sums

Theorem
Let $\displaystyle \sum_{n \mathop \in A} a_n$ and $\displaystyle \sum_{n \mathop \in B} b_n$ be absolutely convergent sequences.

Then:
 * $\displaystyle \left({ \sum_{i \mathop \in A} a_i }\right) \left({ \sum_{j \mathop \in B} b_j }\right) = \sum_{\left({i, j}\right) \mathop \in A \times B} a_i b_j$.

Proof
We have that both series are absolutely convergent.

Thus by Manipulation of Absolutely Convergent Series, it is permitted to expand the product as:


 * $\displaystyle \left({\sum_{i \mathop \in A} a_i }\right) \left({\sum_{j \mathop \in B} b_j }\right) = \sum_{i \mathop \in A} \left({a_i \sum_{j \mathop \in B} b_j}\right)$.

But since $a_i$ is a constant, it may be brought into the summation, to obtain:
 * $\displaystyle \sum_{i \mathop \in A} \sum_{j \mathop \in B} a_i b_j$

Hence the result.