Cartesian product of infinite amount of finite sets bijects to? I have an infinite amount of sets containing two values each.
$A_1 = \{a_1,_1; a_1,_2\}$ ... $A_n = \{a_n,_1; a_n,_2\}$ and so on
If a make a Cartesian product of all these sets $A_1 \times A_2 \times \ldots \times A_n \times \ldots$, 
will the product be a bijection to, or in other words, will the cardinality be equal to that of natural numbers $\mathbb{N}$ or real numbers $\mathbb{R}$?
 Sorry if my question is confusing due to wrong terms.

For concreteness, and without loss of generality,
 let's assume $A_1=A_2=\dots=A_n=\dots$, and label your two elements $0$ and $1$.
What can you say about the set $\prod_{n=1}^\infty A_n$, say compared to $[0,1)$? (E.g., thinking about binary expansion.)
