In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string.

My question is: why can't we begin by representing each natural number as an infinite bit string? So that 0 = 00000000000..., 9 = 1001000000..., 255 = 111111110000000...., and so on.

If we could, then the diagonal argument would imply that there is a natural number not in the natural numbers, which is a contradiction.