I can see that $A$ is, by definition of the product, the set of sequences $(a_1,a_2,...)$ such that each $a_i$ is itself a sequence belonging to $A_i$.
I'm wondering is there another way of looking at this product, possibly involving commutativity of the cartesian product.
I am generalizing my question from a more specific question that is stumping me from a paper but it will be hard for me to give the exact specifics.
EDIT: I meant to mention, is there a way of viewing this product $A$ as a set of sequences of natural numbers?
Well, as you are talking cartesian products. $\prod A_i = \{(a_1,a_2, ..., a_n)|a_i \in A_i\}$ which is a set of all possible $n$-tuples.
It's not a sequence itself but instead if $A_i$ is a sequence in some space $X_i$
$\prod A_i = A_1 \times.... \times A_n \subset X_1 \times ... \times X_n$.
If the sequences are all sequences from the same space $X$ (say, $\mathbb R$) then $\prod A_i \subset X^n$.
So if they are all sequences of real numbers, then $\prod A_i \subset \mathbb R^n$.