elementary set theory - Bijection between SxS and S where S is an infinite string of 1's and 0's

Denote $S = \{(a_1, a_2, a_3, \dots)| a_i \text{is 0 or 1}\}$.

So I know if I think of one S as $\{(a_1,a_2,a_3,\dots)\}$ and another S as $\{(b_1,b_2,b_3,\dots)\}$, I can create a function that spits out something like $\{(a_1,b_1, a_2,b_2, a_3, b_3, \dots)\}$. I've seen this sort of thing before when showing that (0,1)x(0,1) bijects to (0,1), but I'm having trouble proving that such a function is injective and surjective. Thanks.