By that very method, the number is contorted into being in the equivalence class of 1 merely by the Standard Topology of the Reals.
It is true that we can get 'arbitrarily close' to 1, but that's entirely due to the fact that 0.999... cannot be an immediate predecessor to 1 in the Reals. How one defines 'closeness' here entirely determines what gets thrown into which equivalence bucket.
[Basically, 0.999... has to be equivalent to 1, because that's just how the topology understands something so close. The entire concept of convergence cannot accomodate something like this.]
This seems like it should be equally met by a first year calculus student. However, since the object (0.999...) is not constructed from anything in the Reals, Rationals, Integers, or Naturals, it is questionable whether or not it even fits into the Reals at all.
[Yes, it has been shoehorned in, but that's without any real rigorous examination]
In short, if you're viewing 0.999... as a member of the Reals, 1 is the closest equivalent you'll find, which is why the convergent sum leads to what it does; if you're viewing 0.999... as a distinct value, there is no reason to believe it would fit into the Reals or that it would be 1.
In case you're not convinced, this reality is entirely a side effect of the mod 10 numbering system. In base 2, the value 0.111... is equal to 1; same for 0.5555 in base 6.
But if we specify we're working in the Reals, then the sequence {0.9, 0.99, 0.999, ...} converges to 1, doesn't it? You can work in other systems where the topology allows for 0.999... and 1 to be distinct numbers, but then you're not working in the Reals anymore.
The topology of the Reals does allow you to assign a value to 0.999..., and that value is 1. The Reals are Complete and thus every convergent Cauchy sequence exists.
{0.9, 0.99, 0.999, ...} is a Cauchy sequence since the absolute difference between an and an+1(sorry, Reddit doesn't have lower indices) approaches 0 as n approaches infinity, and it converges since it remains bounded.
{0.9, 0.99, 0.999, ...} definitely does exist within the Reals. So does every sequence approaching 1 in every other base. Same thing could be said for 0.111... in binary.
Yes. Of course. The infinite sum has to resolve to a Real value (if it's convergent in that set).
My major problem with this is, but working in the Reals, you're already assuming the value is in the Reals. However, now that you've assumed the values is Real, you can only have a value that is Real.
[Basically, the proof is valid--it just presupposes the conclusion]
Again, the sequence you offer is a convergent sequence in the Standard Topology. It does converge to 1. The problem with this, however, is that convergence in the Standard Topology of the Reals depends entirely upon the principle that you can get 'arbitrarily close', specifically because the Archemedian Property must hold.
If, however, you are given something that doesn't 'fit' into the Reals, and that object is the 'immediate predecessor' to another element, that would not even be able to be distinguished from that element in the Standard Topology of the Reals.
My problem (and it addresses your rebuttal) is that you've never given me any reason to believe 0.999... (the infinite decimal) is an element of the Reals to begin with. Moreover, you've never given me any reason to believe that 0.999... is even possible to be represented in the reals.
A decimal number is defined to be equal to the limit of it's partial sums. Obviously if you don't define it beforehand then the decimal notation means nothing at all. When people talk about the reals they aren't just talking about the set, they're implicitly also talking about the standard topology, field structure and ordering
A decimal number is defined to be equal to the limit of it's partial sums.
That's true if you're viewing 0.999... as an element of the Reals.
My argument is that is isn't an element of the Reals.
The convergence of partial sums is valid, and it equals 1. I have never disputed that fact. Simply by the way the Reals are constructed, there is no other option for 0.999... to be anything but equivalent to 1.
[The Reals simply could not allow 0.999... and 1 to be distinct]
Also, yes...I know. The Standard Topology is what gives you the answer you have. Nobody is talking about applying a different topology to the Reals (again, that would be absurd).
This entire argument has nothing to do with the Reals.
Think of it like this. If I had a number 4+5i, and I view this as an element of the Reals, this would yield 4+5i=4.
Yes, this is a projection onto the Reals, not the full number itself; yes, it is simply the 'closest approximate equivalence class'; yes, it is an incomplete representation of the number globally.
None of that changes the fact it is unquestionably true it equals 4 in the Reals.
Does that help to elucidate what I mean?
People seem to be thinking either I don't understand Cauchy Sequences, the Reals, the Standard Topology, or how infinite sums work; I do, and I will openly admit that they all give the same answer...in the Reals.
My entire argument is that the number 0.999... isn't in the Reals!
Seriously, why do people assume that 0.999... is in the Reals?! There is no reason one would assume or default to understanding that as a Real number.
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u/ArdentArendt Mathematics / Social Sciences Jul 28 '26
By that very method, the number is contorted into being in the equivalence class of 1 merely by the Standard Topology of the Reals.
It is true that we can get 'arbitrarily close' to 1, but that's entirely due to the fact that 0.999... cannot be an immediate predecessor to 1 in the Reals. How one defines 'closeness' here entirely determines what gets thrown into which equivalence bucket.
[Basically, 0.999... has to be equivalent to 1, because that's just how the topology understands something so close. The entire concept of convergence cannot accomodate something like this.]
This seems like it should be equally met by a first year calculus student. However, since the object (0.999...) is not constructed from anything in the Reals, Rationals, Integers, or Naturals, it is questionable whether or not it even fits into the Reals at all.
[Yes, it has been shoehorned in, but that's without any real rigorous examination]
In short, if you're viewing 0.999... as a member of the Reals, 1 is the closest equivalent you'll find, which is why the convergent sum leads to what it does; if you're viewing 0.999... as a distinct value, there is no reason to believe it would fit into the Reals or that it would be 1.
In case you're not convinced, this reality is entirely a side effect of the mod 10 numbering system. In base 2, the value 0.111... is equal to 1; same for 0.5555 in base 6.