The number ...1 in your first notation is actually -1, using 2-adic interpretation. The number 0.1... is actually 1, which lies outside your expected range.
Maybe another base will help. In base 3:
0.0...
0.10...
0.20...
0.010...
Once again, a naive diagonalization yields 0.1... but this time at least we have not extended past 1. Is there a way to avoid that convergence? Sure, we can change the 0 digit to either 1 or 2 randomly, or based on some pattern or enumeration, instead of just 1. So now we can take many diagonals, and they all look like:
Augh! What happened? Our diagonalization appears to have revealed an infinity of missed reals! This is similar to the construction of the Cantor set (https://en.wikipedia.org/wiki/Cantor_set) and hopefully illustrates the problem with your enumeration of the reals.
Maybe another base will help. In base 3:
0.0... 0.10... 0.20... 0.010...
Once again, a naive diagonalization yields 0.1... but this time at least we have not extended past 1. Is there a way to avoid that convergence? Sure, we can change the 0 digit to either 1 or 2 randomly, or based on some pattern or enumeration, instead of just 1. So now we can take many diagonals, and they all look like:
0.121212... 0.122122... 0.121121... 0.22212221... ...
Augh! What happened? Our diagonalization appears to have revealed an infinity of missed reals! This is similar to the construction of the Cantor set (https://en.wikipedia.org/wiki/Cantor_set) and hopefully illustrates the problem with your enumeration of the reals.