u/DawnOnTheEdge

>On Aug. 28, 2014, the tan suit that then-President Barack Obama wore to a White House press conference became an unexpected story as right-wing critics attacked Obama[....]

>Here’s the news: Obama gave the suit away cleaning out his closet.

>So, where is the suit?

>Follow Lynn Sweet, former Sun-Times Washington Bureau Chief, in her quest to find the original tan suit.

“Where you’re never too old to play—but you may well be too young.”

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}, the number greater than or equal to every number in the set and less than or equal to all upper bounds of the set, OK. I accept we’re talking about different things.

Does anybody here disagree,

1. 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 (in this case, under these assumptions) a lowest upper bound doesn’t exist.
2. If we don't have infinitesimals, but we do have all the rational numbers and arithmetic, the lowest upper bound exists and equals 1.
3. So, either the LEB equals 1, or it doesn’t exist, or we can't do arithmetic with it.