If this statement is true "But, you say, there’s a one at the end that string of zeroes. No, there isn’t, because the string of 9s doesn’t end."

Then the same can be said for 0.000000...1 you never reach the 1 at the end therefore you haven't really done the subtraction properly. To put it your way "the string of 0s [before the one] doesn't end."