[BuuQu9hu]

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:

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.

[alok-g]

Thanks. This was helpful. I am not sure that I fully get it, but I think I do. :-)