Normally when I see this form of the product of infinite series:
$$ \left(\sum_{n=0}^\infty a_n\right)\left(\sum_{m=0}^\infty b_m\right) = \sum_{n=0}^\infty\sum_{m=0}^\infty a_nb_m $$
I see the claim that one of the series must be absolutely convergent (e.g. How to Multiply Two Infinite Series Correctly?). However, I have a simple proof that it is true for any convergent series. If this identity is true, it seems like it would be a proposition listed in most real analysis books, but it doesn't appear in any I've checked, which makes me suspect my proof may be invalid.
Here it is:
Suppose $ \sum_{n=0}^\infty a_n $ is a series of reals converging to $ x $, and suppose $ \sum_{m=0}^\infty b_m $ is a series of reals converging to $ y $.
For all $n \in \Bbb{N} $, we find that $ \sum_{m=0}^\infty a_nb_m $ is a convergent series with sum $a_ny$. And of course, $\sum_{n=0}^\infty a_ny$ is a convergent series with sum $xy$. Combining these two facts,
$$ \sum_{n=0}^\infty \sum_{m = 0}^\infty a_nb_m = \sum_{n=0}^\infty a_ny = xy $$
Is this valid? If so, why doesn't this appear in lists of properties of infinite series right alongside $ \sum_{n=0}^\infty a_n + \sum_{n=0}^\infty b_n = \sum_{n=0}^\infty (a_n + b_n)?$