1. Originally Posted by EVOEx Just when I thought we agreed :P. My proof used limits and I never assumed that "1 = .999...". It only uses the definition of "0.999..." and the limit of "10^-x" as x goes to infinity. If you really think that, can you point out where in my proof I assumed any such thing (just out of curiousity).
Your proof equalizes an infinitesimal to 0 based on the established rule that, there not being a distance between any two numbers, those two numbers are the same. That's the genesis of the proof. Sang-drax proof just above me uses that rule to prove that 0.999... equals 1 without that requiring establishing the identity of an infinitesimal. Essentially you both prove the same thing though, by slightly different routes.

However I find this argument curious (emphasis mine): Originally Posted by MacNilly
However, if the error (epsilon) is _arbitrarily small_, and if the function under consideration (0.999... in this case) has a maximum upper bound, we can achieve a value _arbitrarily close_ to that upper bound. Thus, the function is effectively equal to that upper bound (limit). That is, there is no number one can name between 0.999... and 1 (no matter how many 9's you choose, I choose one more 9).
But isn't that also an argument that contradicts the whole thing? I can reword the last sentence to an equivalent sentence that puts things in a different perspective: "There is always a real number between 0.999... and 1 (no matter how many 9's you choose, I choose one more 9)."

This is at the core of the concept of infinity. Where you wish to establish you know what happens when we reach infinity, I wish to establish we actually don't. Where you wish to establish you can prove it, I wish to establish you can't. And my own proof for that theory is exactly the contradiction that the proofs we have seen so far present when we consider that for any real number we can't think of, 0.999... is never equal to 1.

This is the basis of my thesis that you can't prove 0.999... equals 1. You can only establish it through an unproven rule that serves the purposes of limits and that gives consistency to the Real number system (that unproven rule being that there is no distance between 0.999... and 1).

I can certainly accept the ruling. But cannot accept that as a proof. 2. Originally Posted by Mario F. I can certainly accept the ruling. But cannot accept that as a proof.
hmm, remind me again why the following isn't good enough to stand as a proof

Code:
```1/9 = 0.(1)
9*(1/9) = 9*0.(1) = 0.(9)
9/9 = 1 = 0.(9)``` 3. Originally Posted by EVOEx Just when I thought we agreed :P. My proof used limits and I never assumed that "1 = .999...". It only uses the definition of "0.999..." and the limit of "10^-x" as x goes to infinity. If you really think that, can you point out where in my proof I assumed any such thing (just out of curiousity).
I was wrong. The two(lim and sup) are, in fact, not similar, in this case(I was thinking about limits of sequences, which are similar to supremums of subsets of ordered sets. In other words, I was WAY off topic.). I would explain my issue with sup, but it would be an necessary extension to an unnecessarily long thread. (That is, unless you want an explanation.)

After 11+ pages, I'm ready for this thread to just die.  4. Originally Posted by Mario F. Your proof equalizes an infinitesimal to 0 based on the established rule that, there not being a distance between any two numbers, those two numbers are the same.
I agree with "User Name:" that this thread is ready to die, as everyone but Mario agrees about the proofs I believe.

Let me just try it one more time for you. My proof never uses that infinitesimal equals 0, my proof uses that the LIMIT of something GOING TO infinitesimal equals 0. The two are completely different, and the latter is easily the definition of the limit.
Apparently you fail to see the difference, even with several people explaining and even copying the definitions of limits. And yet you keep insisting on this matter, which I don't get. You're too smart, I believe from all your posts, not to understand it. But for some reason you choose to ignore it.
What is it, some limit-phobia? 5. Originally Posted by EVOEx Let me just try it one more time for you. My proof never uses that infinitesimal equals 0, my proof uses that the LIMIT of something GOING TO infinitesimal equals 0. The two are completely different, and the latter is easily the definition of the limit.
Apparently you fail to see the difference, even with several people explaining and even copying the definitions of limits. And yet you keep insisting on this matter, which I don't get. You're too smart, I believe from all your posts, not to understand it. But for some reason you choose to ignore it.
What is it, some limit-phobia?
Maybe you should read my arguments better. Because it's all there. I've explained to you numerous times the problem that what you are saying and what you are actually doing with your proof are not the same thing.

I'll dissect this quote of yours so that once and for all you understand my argument with your proof. And do read the whole thing this time. I'm tired of repeating myself.

>> My proof never uses that infinitesimal equals 0, my proof uses that the LIMIT of something GOING TO infinitesimal equals 0.

Very true. Your proof returns 0 as the L of an infinitesimal. No complains here.

>> The two are completely different, and the latter is easily the definition of the limit.

Very true. They are completely different. Your own words. Completely. Completely different. Again, for it to really sink: Completely different. And I agree. They are completely different. An infinitesimal does not equal zero. The limit of something going to it does.

>> Apparently you fail to see the difference, even with several people explaining and even copying the definitions of limits.

No. I see the difference. There's a difference. They are completely different. I've always said they are different. That's the basis of my whole argumentation. But do YOU see the difference? I doubt it. Because your proof consists of this: Originally Posted by EVOEx
lim(x -> inf) ( 1*10^-x ) = 0

So:
1 - 0.9999... = 0
1 = 0.9999...
If an infinitesimal does not equal zero as you say, how come you bring it to algebra as being 0 in the final part of your proof? It's either different or equal to 0. You have to make up your mind. Because you don't translate the meaning of the equality sign on limits into plain algebra without that resulting into you turning L into not just the limit of zero, but actually 0.

>> And yet you keep insisting on this matter, which I don't get. You're too smart, I believe from all your posts, not to understand it. But for some reason you choose to ignore it.

As I don't get why do you insist in saying that you are not making an infinitesimal equal to 0 when your proof concludes exactly with that. Moreover, how can you claim that 0.999... is 1 without that invariably meaning that an infinitesimal is 0.

More importantly, if you claim that an infinitesimal is not zero (but the limit of something going towards it is what equals 0) then we can conclude that 0.999... does not equal 1, but the limit of something going towards 1 is what equals 1. The only thing you proved was that 0.999... tends towards 1. Not that it equals 1.

Let me say this to you again... you cannot prove the identity of a real with non-terminating decimals. You can only establish its limit.

>> What is it, some limit-phobia?

I've had the opportunity to say a few times (you probably didn't read them, as you probably won't this post) that I have no issues with calculus. Only with the fact people insist they can prove 0.999... equals 1. They can't. They can at best establish why it is useful such a rule to exist within the domain of R (which is what I defend you did and did well). But otherwise that proof is impossible.

More so because the different number systems all have axiomatic consistency. You don't have an axiom that is true for one number system but that proves it false on one that extends it. Instead what you have are certain unspoken rules, universal agreements as to how to represent certain quantities within a limited number system incapable of properly representing them. Not much different from having to be stuck with only the Natural numbers and defining a approximation rule to represent reals. You cannot prove that 1.5 is 2. You just establish it.

Finally one quick explanatory note:
>> even with several people explaining

I suppose you take me for a sensible person? You don't expect me to be insane at least. I'd need to be cuckoo to pursue this debate so long and against the opinions of so many people if I had seen the validity of yours and their arguments, won't you agree? But I'm not crazy. So something else is afoot here.

Besides if you believed I was crazy, you would have already stopped replying. Which you haven't. So I believe you either a) see room for debate, or b) don't like the fact someone is negating the validity of your proof.

I want to believe it's "a)". So with that in mind can you accept that perhaps, just perhaps, no one has actually provided anything concrete, palpable, and unquestionable, that refuted all I have been saying on this post?

Because I also want to believe you know I am man enough to immediately concede and put aside all my arguments. The fact no one did that, the fact no single so-called "proof" i have seen on this thread doesn't immediately raise a number of questions for a cynical (but also unruly and always inquisitive) mind like mine, is what drives me. Not into trying to annoy or troll you and everyone else, but into trying to understand and learn.

Knowledge is what I seek. But don't expect me to accept certain things just because "everyone says it is like so", when I find flaws on them. I will never accept those things until I'm convinced there are no flaws (just my poor judgment). Until then, tough luck; I won't budge. And make no mistake, I question myself all the time. So it's not just you trying to convince me. It's me too. I don't produce convictions and defend them. I attack them, perhaps more violently than you think. Because there's little pleasure with arguing against 10 people for 2 weeks and 13 pages. 6. Originally Posted by Mario F. If an infinitesimal does not equal zero as you say, how come you bring it to algebra as being 0 in the final part of your proof? It's either different or equal to 0. You have to make up your mind. Because you don't translate the meaning of the equality sign on limits into plain algebra without that resulting into you turning L into not just the limit of zero, but actually 0.
And this seems to be the problem, here, saying the meaning of the equal sign on limits is different. The equals sign never means anything other than "is equal to", even if a limit is involved:
Code:
`lim[x -> infinity] 10^-x = 0`
We can even put the limit in a variable:
Code:
`t = lim[x -> infinity] 10^-x`
And then all the following is true:
Code:
```t = lim[x -> infinity] 1/x = lim[x -> infinity] -1/x
t = 0
t*5 = 0
t + 1 = 1```
Because t is actually 0. Equal to. There is no special meaning to that. And one of the basic rules of equality (again, no matter if limits are involved) says:
Code:
```If:
a = b
b = c
Then:
a = c```
In my proof, I "showed" (the latter I didn't actually show, but those are basic identities) that:
Code:
```1 - 0.9999... = lim(x -> inf) ( 1*10^-x )
lim(x -> inf) ( 1*10^-x ) = 0```
And again, the equals sign for the limits has exactly the same meaning as for anything else, so we can establish from the basic rules of equality that:
Code:
```1 - 0.9999... = lim(x -> inf) ( 1*10^-x ) = 0
1 - 0.9999... = 0```
Why am I still arguing? Because I'm hoping I'm missing something, that I'm hoping you'll show me something I have never known. Or maybe the other way around. Though I'm still convinced I'm right. 7. Originally Posted by EVOEx And this seems to be the problem, here, saying the meaning of the equal sign on limits is different. The equals sign never means anything other than "is equal to", even if a limit is involved
So saying that the limit of 1 - 0.999... equals 0 is the same as saying that 1 - 0.999... equals 0?

You also said:
My proof never uses that infinitesimal equals 0, my proof uses that the LIMIT of something GOING TO infinitesimal equals 0. The two are completely different, and the latter is easily the definition of the limit.

So... help me here.... 8. Originally Posted by Mario F. So saying that the limit of 1 - 0.999... equals 0 is the same as saying that 1 - 0.999... equals 0?
Where in my proof do you see the "1 - 0.9999..." part having a limit? The only assumption I make is that "0.9999..." is equal to:
Code:
`0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )`
That is, the limit of the number as you add more and more 9's. Then:
Code:
```1 - 0.9999... = 1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
1 - 0.9999... = lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )```
(no limit in that "1 - 0.9999...", though yes, it's equal to a limit)

Then I show that:
Code:
`lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = 0`
Hence:
Code:
```1 - 0.9999... = lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x ) = 0
1 - 0.9999... = 0```
There's never a limit any place I don't write it. I didn't even use implicit limits to avoid confusion. 9. 1)
I want to generalize to the point that whenever you use infinity you have to either prove that the process is valid or define it for infinity.

For example:
Code:
`0.9999.... + 0.9.....`
To calculate the sum you have to use a very specific method. In this case you would see that it is hard or impossible to sum to decimals with infinite number of decimal digits the way sum is defined.

The same for things like
Code:
`0.999... * 10`
Conclusion: the important part is to prove the sum or multiplication. Everything else is pure mathematics...

2)
When using sums (Σ) or limits there are some definitions. One definition used by EVOex is that the sum of infinite number of parts is its limit. Its the same thing. So by definition
Code:
`Σ(x=1...inf)(9*10^-x) = 1`
Using limits on the above as pointed out leaves you at one point with
Code:
`lim(...)(0.999... )`
which again is not defined directly. You can assume that 0.999... is a number and you go to the field of comment 1) basically.

---------
Concluding, you need to add some rules for infinite decimal digits on a decimal number. Meaning more axioms. To clarify this, you can prove that if "1+1=2" then "1+2=3" with something that everybody would accept. If you try to do something similar involving infinity you would see that not everybody will accept it and you would have disagreements on axioms. Which you cannot prove further by definition.

If somebody posts those rules and the person agrees with them all, then you can go from there and prove your point. So the agreeing and disagreeing is done with conditions that are not clearly stated. If A disagrees and B agrees this means that A might accept some axioms that B doesn't. Or putting this better, A might agree IF the axioms are stated as true.

For me those conditions, those axioms are the interesting part. 10. I don't remember any axioms or theorems being mentioned to support a dissenting view. But then again, I didn't read the whole thread. 11. Originally Posted by Mario F. Your proof equalizes an infinitesimal to 0 based on the established rule that, there not being a distance between any two numbers, those two numbers are the same. That's the genesis of the proof. Sang-drax proof just above me uses that rule to prove that 0.999... equals 1 without that requiring establishing the identity of an infinitesimal. Essentially you both prove the same thing though, by slightly different routes.
That rule is obviously correct and although I don't use it per se, I use properties of the real numbers to ensure that the limit exists.

0.999... = sum(k=1...oo)9 * 10^(-k) = 1.

is a perfectly valid proof. Here, "oo" should of course not be thought of as a number (it's not), but rather as a convenient form of notation with a well-defined meaning. 12. Originally Posted by C_ntua When using sums (Σ) or limits there are some definitions. One definition used by EVOex is that the sum of infinite number of parts is its limit. Its the same thing. So by definition
Code:
`Σ(x=1...inf)(9*10^-x) = 1`
Using limits on the above as pointed out leaves you at one point with
Code:
`lim(...)(0.999... )`
which again is not defined directly. You can assume that 0.999... is a number and you go to the field of comment 1) basically.
Acctually, I think you're wrong. See, this:
Code:
`Σ(x=1...inf)(9*10^-x)`
As I said before, contains an implicit limit. That is to say, it is equal:
Code:
`lim[t->inf] Σ(x=1...t)(9*10^-x)`
Per definition. I know it's not going to be considered a reliable source, but maybe you believe me if I post a link stating that on wikipedia:
Limit of a function - Wikipedia, the free encyclopedia
See the first in that section. It states that writing the sum of something to infinity is actually saying the limit of that. It's quite obvious, as you can't actually SUM an infinite amount of things, so saying the sum of an infinite amount of non-zero numbers doesn't make sense by itself, which is why we all agreed it's an implicit limit. 13. ## Code:
`0.999... = lim[t->inf] Σ(x=1...t)( 9*10^-x )`
which then will have the argument of differentiating a limit from an actual number.
Meaning that
Code:
`x->10`
and
Code:
`lim[x->10](x) = 10`
are not the same.

So that is why I say
Code:
`0.999... = Σ(x=1...oo)( 9*10^-x )`
because I am implying that oo is truly infinity. A well defined meaning, but you cannot calculate this with "classical" processes. If you use a limit, you are simply changing the meaning of infinity, slightly, but still changing the meaning.
What the article in wikipedia says is that
Code:
`Σ(x=1...oo)( 9*10^-x ) IS  lim[t->inf] Σ(x=1...t)( 9*10^-x )`
not that they are equal, meaning that the meaning of both under the theory of limits is the second. That exactly because you cannot have an infinite SUM, the closest thing is that you have a sum of parts that are NOT infinite, the are t where t->oo. This is certainly not true with 0.999... where it DOES have infinite parts.

If you accept that 0.999... is actually a number, you establish that, you accept that it has a specific quantity then that is a different story. Because a limit of a number is the number itself. But you are just saying that
Code:
`0.9999 IS 1.0`
that there is actually no difference. It is just a different way to represent it. Yes, you can use the equal (=) sign instead of IS, that is not the point. The point is that unless you define 0.999... to have a specific quantity, not use it as a process where you have to calculate the resulting quantity, then you cannot prove that it is equal to 1.

So if 0.999... is a process, you cannot prove it is 1.
If 0.999... is a number, it is 1, not <1 or >1 but exactly 1.

In th end, since 0.999... is used as a number, you would have to expand the real numbers. When talking about numbers, you have to think of order and difference. Thus the distance needs to be defined.

The thing is that
Code:
`0.333... - 0.222....`
is something that you can define. You cannot compare all digits, because they are infinite. You cannot compare two infinite things, that is a very fundamental thing about infinity. What you can do is compare infinity with a number. So you can do
Code:
```0.34 > 0.3333......
0.23 > 0.2222......
0.34 - 0.23 > 0.333... - 0.22222
0 > 0.333.... - 0.2222
0.333.... > 0.2222....```
for the first two you just add 0.01 > 0, thus they are true.

Now if you use a similar logic for 0.999.... and 1 you won't succeed. Because you cannot find a decimal number that is greater than 0.999.... and smaller than 1.0. In the end the only way to actually compare 1 and 0.999... is to use a theorem or an axiom. Or just leave 0.9999 undefined or something you cannot really use. Doesn't matter if you can express it or write it, unless you can actually compare it with 1.0 you cannot use it as a number.
This is solved by using a theorem or an axiom which basically will but "trailing 9s" as the next decimal digit being 1. You will just define it like that. You cannot prove that this is not true either so there is absolutely nothing wrong about this.
You can still use 0.999... as a number and prove that "0.3333... * 3 = 0.999... = 1" for example and have a consistence theory. But unless you use an axiom like this I really don't see a proof here.

To be clear this:
Code:
`0.999... < 1`
is not something you can prove either. Unless you assume that 0.999... is not a number, but a process for the same reasons.

So my conclusion again is that it depends how you see 0.999.... A process or a limit? I would stick with this as the answer and any proof should use one of the two. 14. Originally Posted by C_ntua Code:
```0.34 > 0.3333......
0.23 > 0.2222......
0.34 - 0.23 > 0.333... - 0.22222
0 > 0.333.... - 0.2222
0.333.... > 0.2222....```
0.34 > 0.3...
0.23 > 0.2...

I can agree with the above. I cannot agree with anything you said following. Just because a > a0, and b > b0, it does not imply that a - b > a0 - b0. We could try doing some of what examples you have:

0.34 - 0.23 = 0.11
0.3... - 0.2... = 0.1...

Or if you prefer:

34/100 - 23/100 = 11/100
3/9 - 2/9 = 1/9

That's right, isn't it?

0.11 is not greater than 0.1... because 11/100 > 1/9 is contradictory. In the decimal expansion, 0.11 has 0 for the hundredths place, while 0.1... has 1 for that, and all other places. Following through, that means that 0.34 - 0.23 > 0.3... - 0.2... is a false statement. Plus, you've made another mistake. There are numbers between 0.2... and 0.3... 2/9 != 1/3 and any real in between those, like say 0.235, is proof that they aren't the same number. However, there may be no real in between between say 0.23 and 0.229 with 9 recurring, unless you can think of one.

So yeah your whole argument just does not work. You're not wrong because my axioms are different from your axioms either. You're wrong because no one said what you're trying to prove. And I can only guess what that is. Maybe that all non-terminating decimal reals are not equal to their nearest neighbor for some intuitive meaning of "nearest". In fact it's because they have neighbors that they aren't equal. Rather, if 0.9... is equal to 1, then you could write 0.9... as 1, and that has implications for many other reals. 0.2 could be 0.19 with 9 recurring and so on. It's just a different way to write the same number. That is to say that these numbers are equal, which in itself is a comparison.

The reason you can accept various proofs: If you're working with what you think is 0.9... and you get 1 as a result, as long as there is no error in the process equations, then they are equal. And just because you can prove 0.9... = 1 with process equations, it does not make 0.9... a process equation. 0.9... is a number, just like Sang-drax said. 0.9... has a value, (whatever it may be,) and it comes with no operations. Terminating decimals themselves have infinite expansions as well, with 0 filled in for any places after a given place. We do not argue their status as numbers. 15. What is a process equation? Popular pages Recent additions 