Re: Doubly Infinite Strings.

I am constructing a doubly infinite string using the following process: I start with the string 0101. In each step, I insert a 0 to the left of each 1 in the string. After the first step, the string becomes 001001. After the second step, it becomes 00010001, and so on.
 With each iteration, the string grows longer, but only in one direction.
 If I continue this process forever:
      To the left of the first 1, there will eventually be an infinite number of 0s.
     To the right of the first 1, there will also be an infinite number of 0s, followed by another 1.
 Does this process create a valid doubly infinite string?