For a sequence $(a_i)_{i \in \mathbb{Z}}$ of real numbers we consider the two sided sum: $$\sum_{i=-\infty}^\infty a_i$$
We say that this sum is (absolutely) convergent, if both parts: $$\sum_{i=0}^\infty a_i,\sum_{i=1}^\infty a_{-i}$$ are (absolutely) convergent and define:$$\sum_{i=-\infty}^\infty a_i := \sum_{i=0}^\infty a_i + \sum_{i=1}^\infty a_{-i}$$Now consider two sequences $(a_i)_{i \in \mathbb{Z}}, (b_i)_{i \in \mathbb{Z}}$ such thattheir two sided serieses both are absolutely convergent. I want to prove, that the series defined by the sequence:$$c_n := \sum_{j = -\infty}^\infty a_j b_{n-j}$$is also absolutely convergent, and that it holds:$$\sum_{i=-\infty}^\infty c_i = \sum_{i=-\infty}^\infty a_i \cdot \sum_{i=-\infty}^\infty b_i$$
I cannot find this statement anywhere, and since I did not manage to prove it, I am not sure if it holds. I did manage to prove the part that $\sum_{i=-\infty}^\infty c_i$ is absolutely convergent, however I have failed at proving the equality.
I would be thankful for help,Greetings