Does anyone here disagree that either the LEB of {0.9, 0.99, 0.999, ...} is 1, or it doesn’t exist, or arithmetic doesn’t work?

Take the set {0, 0.9, 0.99, 0.999, ...}. Or, more formally, {1-10⁻ⁱ | i∈ℕ}. (You might or might not include 0.) For any natural number, zero-decimal-point-that-many-nines is in the set.

If you want to say what-you-call-0.(9) is something not equivalent to sup{1-10⁻ⁱ | i∈ℕ}, OK. I accept we’re talking about different things.

Does anybody here disagree,

1. This is a bounded set of rational numbers. If our number system has no infinitesimals, its lowest upper bound exists and equals 1.

2. If our number system has some infinitesimal positive number ε, greater than 0 but smaller than any 10⁻ⁱ, and we try to define the lowest upper bound of the set as 1-ε, and we can add, multiply and compare infinitesimals, we find that 1-2ε is a lower upper bound than the lowest upper bound. But that’s a contradiction! So a lowest upper bound doesn’t exist.

3. So, either the LEB equals 1, or it doesn’t exist, or we can't do arithmetic with it.