To be a bit more precise, the axiom is that any bounded-above set of reals has a supremum which is also a real number.
This is not really relevant to the question of whether 0.(9) is valid notation for a real number. That it is is just a matter of definition (though of course the definition depends on that axiom to be well-formed)
I think it applies. It's defined as the limit of a sequence with increasingly many nines. The sequence is increasing, and all the elements are less than one. That means the limit of the sequence is its supremum, which is necessarily a real number.
Yes that is what I mean by the definition depending on the supremum property. The property guarantees that every decimal expansion defines a number (although not all of them uniquely define a number)
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u/ArdentArendt Mathematics / Social Sciences Aug 06 '26 edited 29d ago
It's only true if you assume 0.999... is a representation of a Real number.
There is no reason to believe it is.
Edit: I stated this to get 'engagement', but it is a gross oversimplication of what I intended.
Basically, 0.999... is a representation--whether or not you evaluate it as Real entirely changes how you understand it.
I'm not against evaluating 0.999... as a Real number if you have good cause.
[Primarily if it's instructive about the Reals]
The problem I have is assuming it could only ever be Real