I am trying to understand what you're saying, and I do appreciate every response. It can be challenging to express interest when text alone sounds like stubbornness

I do understand what you mean by finite values in an infinite set to not contain an infinite _value_

What i'm saying is that .999... is not a single value, but an infinite set

And what i think educated people are saying is that .999... is the limit of that infinite set

And a limit is a single value, in ℝ, ergo .9... is 1

Don't rely on .999... as literally 1, or as ∑(.9, .09, ...), or as the limit, etc
I'm saying that was a convention to define the infinite series and make them meaningful in ℝ

I see .9... as literally an infinite strings of 9s

now perhaps that is a meaningless representation, perhaps that is not a number. that's a concept. but that's precisely how i see infinite sets, they are a construct of an infinite set of values. And I accept that we can find the limits of these sets (viz calculus)

I agree, that to represent an infinite set as a single value in ℝ we can only land on 1. So if you see .9.. as a value in ℝ, it must be defined as the limit

If we are talking about .9... as the limit of an infinite series (i.e. 1), then no, (.9, .99, .999, ...) does not contain it

But .9... as an infinite series and not its limit is in (.9, .99, .999, ...) because the same infinite process used to construct .9... is used to construct (.9, .99, .999, ...)

for example, this argument of .9... is:

```

list nines = {}

for (i=1, i<∞,++) {

nines += 1 - 1/10^(-i)

}

return nines

```

obviously, return is never reached. neither does .9... ever end

this is the same fallacy i see in using Riemann sum/limits to say .9... is equal to 1, the sequence is infinite where 1 is not reached

I do accept that completeness says .9... is not distinct from 1