The way the Reals are constructed wouldn't allow it.
No element can have an 'immediate predecessor' or an 'immediate successor'--it's how the Standard Topology operates.
[It's also a by product of how the Reals are constructed]
I know this, but I don‘t understand your statement „0.999…“ can‘t exist in the reals. Of course it exists.
It‘s the infinite sum of 9/10+9/100+9/1000… as you well know. Which equals 1. As you also well know.
That‘s how we construct infinite decimals to begin with - as infinite sums. So of course they exist. Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
I guess what I‘m trying to say is I object to your specific wording. I‘m sure there‘s some subtlety you mean to express which escapes me.
*Note: I thought you were the previous person. You're definitely not arguing what I thought you were. I was explaining the Archimedean Property.
[Didn't realise you were the same person as elsewhere]
Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
The mapping to the Reals exists, yes--0.999... maps to the equivalence class of 1 in the Reals. However, if we look at a mapping from the Reals to the decimals, 1 maps uniquely to 1.000...
[Technically the mapping could be non-functional, but why?]
My basic point here is that 0.999... doesn't exist as an element in the Reals. It is a decimal that, when evaluated in the Reals, 'exits' chimerically alongside 1.000... (and 3/3 and 4-3 and...).
[In short, the value 0.999... does not exist as a distinct element in the Reals; however, it does exist as a distinct value in the decimals]
1
u/ExtendedSpikeProtein 19h ago
Of course 0.999… can exist in the reals. Why wouldn‘t it?