Alice: 1 = 0.999... Bob: No, they’re different. Alice: Okay, if they’re different then why do you get zero if you subtract one from the other? Bob: You don’t, you get 0.000...0001. Alice: How many zeros are there? Bob: An infinite number of them. Then after the last zero, there’s a one.
Alice is right (as far as real numbers go) but at this point in the discussion
she has not yet proved her case; she needs to argue to Bob that he shouldn’t
use the concept “the last thing in an infinite sequence” (or that if he does
use it he needs to define it more rigorously).
If I may, let me agree with you in dialogue form:
Alice: 1 = 0.999...
Bob: No, they’re different.
Alice: Okay, if they’re different then why do you get zero if you subtract one from the other?
Bob: You don’t, you get 0.000...0001.
Alice: How many zeros are there?
Bob: An infinite number of them. Then after the last zero, there’s a one.
Alice is right (as far as real numbers go) but at this point in the discussion she has not yet proved her case; she needs to argue to Bob that he shouldn’t use the concept “the last thing in an infinite sequence” (or that if he does use it he needs to define it more rigorously).
There is no “after the last” zero.