Problem Statement: Given and define the product to be where . Suppose that converges to absolutely and converges to absolutely. Prove that converges to a value absolutely.
Proof: Since and converge absolutely we know that their product converges to (this is by Rudin Theorem 3.50 which requires only one of the sums converge absolutely in order to guarantee that their product will converge to . This theorem does not guarantee that the product converges absolutely.).
Define where is as defined previously. Also define where . Similarly let . Then it follows that :
Recall that we are trying to show that the sequence of converges to . So let . Since we know that it follows that the limit of the above equation will tend to iff . Let us define (we know that exists since we know the series converges absolutely).
First note that as it follows that since . So let . Then there is a such that for it follows that . But by our definition of this also means that for it follows that .
Now consider :
Now let . Since is fixed the above inequality holds as we take the limit as . But we know that since the series converges and so as it follows that . This implies that , but since may be made arbitrarily small, it follows that .
Thus, we may conclude that and so converges, thus, converges absolutely.
Reflection: The key to this proof is that both original series are absolutely convergent. The necessity of this isn’t immediately clear, but once you try to show that you see that you need to know that you can bound the absolute values of the terms from the series.