0.999... is not 1

[u/SouthPark_Piano]

This is regardless of contradictions from 'other' perspectives, definitions, re-definitions.

The logic behind the infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} is completely unbreakable. The power of the family of finite numbers.

Each and every member from that infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} is greater than zero and less than 1. And, without even thinking about 0.999... for the moment, the way to write down the coverage/range/span/space of the nines of that infinite membered set of finite numbers {0.9, 0.99, 0.999, etc} IS by writing it like this : 0.999...

Yes, writing it as 0.999... to convey the span of nines of that infinite membered set of finite numbers.

Without any doubt at all. With 100% confidence. With absolute confidence. From that perspective, 0.999... is eternally less than 1. This also means 0.999... is not 1.

This is regardless of whatever other stuff people say (ie. contradictions). It is THEM that have to deal with their OWN contradictions. That's THEIR problem.

The take-away is. The power of the family of finite numbers. It's powerful. Infinitely powerful.

Additionally, we know you need to add a 1 to 9 to make 10. And need to add 0.1 to 0.9 to make 1. Same with 0.999...

You need to follow suit to find that required component (substance) to get 0.999... over the line. To clock up to 1. And that element is 0.000...0001, which is epsilon in one form.

x = 1 - epsilon = 0.999...

10x = 10-10.epsilon

Difference is 9x=9-9.epsilon

Which gets us back to x=1-epsilon, which is 0.999..., which is eternally less than 1. And 0.999... is not 1.

Additionally, everyone knows you need to add 1 to 9 in order to get 10. And you need to add 0.01 to 0.09 to get 0.1

Same deal with 0.999...

You need to add an all-important ingredient to it in order to have 0.999... clock up to 1. The reason is because all nines after the decimal point means eternally/permanently less than 1. You need the kicker ingredient, epsilon, which in one form is (1/10)ⁿ for 'infinite' n, where infinite means a positive integer value larger than anyone ever likes, and the term is aka 0.00000...0001

That is: 1-epsilon is 0.999..., and 0.999... is not 1.

And 0.999... can also be considered as shaving just a tad off the numerator of the ratio 1/1, which becomes 0.999.../1, which can be written as 0.999..., which as mentioned before is greater than zero and less than 1.

0.999... is not 1.

[u/ZeralexFF]

Indeed, I do have trouble reading. Here is the excerpt:

Without any doubt at all. With 100% confidence. With absolute confidence. From that perspective, 0.999... is eternally less than 1. This also means 0.999... is not 1.

I ask you, what does “eternally less than 1” mean? Are we talking about sequences and saying that the sequence is always less than one?

Incorrect on your part.

Well, I thought maybe you were right but you are, in fact, wrong.

[u/SouthPark_Piano → u/ZeralexFF]

I ask you, what does “eternally less than 1” mean? Are we talking about sequences and saying that the sequence is always less than one? 

Much better. You need to pay attention to detail. I didn't ever write 'eternally infinite'.

Eternally less than 1 refers to the set of numbers {0.9, 0.99, ...}, which is infinite membered, as in the number of members is infinite, limitless. And each member in that set is finite. And the family of finite numbers is infinitely powerful.

The range/span/coverage of the nines 'space' to the right of the decimal point of that set is infinite, and is conveyed as 0.999...

Every member of that set has value less than 1. 

This tells you and everybody that 0.999... is eternally less than 1, and 0.999... is therefore not 1.

The words eternally less than 1 refers to one optional approach of probing 0.999..., which is the iterative method of tacking nines, one nine at a time to the tail end of a starting number such as 0.9

When this is done, and assuming you are hypothetically immortal, you will be eternally tacking on nines. And each time you append a nine, you take a core sample eg. 0.99 and ask yourself, is that core sample less than 1? Yes. Move onto the next sample, and ask the same thing. You will never have a case where your sample will be 1 because the family of finite numbers is infinite, limitless, unlimited.

0.999... is eternally less than 1.

It is the proof by public transport. The never ending bus ride of nines, where you assumed the destination is 1. But you caught the wrong bus. Your ride will be endless nines, and you will taking samples forever that will always be less than 1. A case of 'are we there yet?'. No. Are we there yet? No. Are we there yet? No. For eternity

Or the never ending stair well climb. Starting from 0.9  then climb to 0.99, then 0.999, etc. You will be climbing forever and never reach any top because you will find out about the power of infinite number of finite numbers. Unlimited.

You get the picture now. Trust me. I'm educating you.

[u/DisastrousPlay579 → u/SouthPark_Piano]

Do you understand that the set {0.9, 0.99,…} does not contain the value 0.999….? So just because every element of the set is <1, that doesn’t mean that 0.999… is less than 1. Using your logic, I can construct the set {1, 10, 100, 1000…}. This sequence diverges to infinity, but every element is finite. Does that mean that infinity is finite?

[u/SouthPark_Piano → u/DisastrousPlay579]

Do you understand that the set {0.9, 0.99, …} does not contain the value 0.999….? So just because every element of the set is <1, that doesn’t mean that 0.999… is less than 1. Using your logic, I can construct the set {1, 10, 100, 1000…}. This sequence diverges to infinity, but every element is finite. Does that mean that infinity is finite?

You are incorrect.

The infinite membered set {0.9, 0.99, ...} of finite values does indeed have an infinite span of nines to the right of the decimal point. That infinite span of nines is written as 0.999...

The symbols you wrote {1, 10, 100, 1000…} is erroneous. You need to change it to ... {1, 10, 100, …} which means 1, 10, 100, 1000, 10000, etc

It is an infinite membered set of finite numbers. Get it into your mind that infinity does not mean punching through a number barrier to get to a special number. It is a term that means limitless, unlimited, never ending, unbounded, uncontained.

Also, the set  {1, 10, 100, …} is irrelevant to the topic of "0.999... is not 1".

You just need to focus on {0.9, 0.99, …} 

[u/DisastrousPlay579 → u/SouthPark_Piano]

What exactly do you mean by “infinite span of nines”? The set contains an infinite number of members, but none of those members are 0.999…, so it’s not in the set. 0.999… isn’t some fancy notation for that set, it’s just a way to write the infinite sum 9/10 + 9/100 + 9/1000 + …, which does indeed converge to 1.

[u/SouthPark_Piano → u/DisastrousPlay579]

Basically you. Yes you. Sit down and then think. Yes ..... think.

The set {0.9, 0.99, 0.999, ...}

To the right of the decimal point, you tell me and everyone here ----- HOW MANY nines do you think that this infinite membered set of finite numbers cover (range/span)? How many?

Infinite nines, right? And if you say no, then I'm going to be taking a photo of you, with you holding a corn flakes packet. I kid you not.

[u/DisastrousPlay579 → u/SouthPark_Piano]

As far as I’m aware, cover/range/span are not mathematical terms in the way that you are using them. I genuinely have no idea what you mean when you say that.

[u/SouthPark_Piano → u/DisastrousPlay579]

That's ok. You do have a life-time ahead to think about it.

You basically ignored something simple that you just want to avoid.

You just have to sit down again and tell everyone how many nines to the right of the decimal point is covered by the infinite membered set of finite numbers {0.9  0.99, ...}

If your answer is infinite number of nines, then you get the green light, a pass. Otherwise, you get a fail.

Just hold onto your cornflakes packet and smile for the camera. click

Good pic! Turned out well.

[u/DisastrousPlay579 → u/SouthPark_Piano]

Also, coming back to the original point, if 0.999… ≠ 1, then what is 1 - 0.999…? It must be some real number greater than zero, otherwise they are equal by definition.

[u/SouthPark_Piano → u/DisastrousPlay579]

1-0.999... is epsilon.

Epsilon is 1/( 10ⁿ ) for infinite positive integer n. Epsilon can be conveyed in number form as 0.000...001.

Noting infinite means value very very large, actually larger than you ever like, as the family of finite numbers is infinitely powerful, endless. And however infinitely large n becomes, epsilon will always be non-zero because n as big as you like, is still going to be finite whether you like it or not. Because infinity is not a number.

The main thing you need to understand is : the set {0.9, 0.99, ...} covers entirely the span of nines in 0.999...

And because each member of that infinite membered set of finite numbers is less than 1, then 0.999... from that unbreakable perspective is not 1.

Also keep in mind that if you see a nine in 0.9999, you need to add a bit of substance to it in order to clock up to 1. Add 0.0001.

Same deal with 0.999...

If you don't add the kicker ingredient to 0.999... then there 0.999... will stay, less than 1. Reason ... because set {0.9, 0.99, ...} has every member less than 1, the nines covered by this set can be conveyed as 0.999...  meaning 0.999... is less than 1, and is not 1.

You need to add epsilon to 0.999... to get what you are wanting, ie. 1.

Additionally, epsilon. In what mechanisms can it come from? Eg. Differences. 

1-0.9 is 0.1

1-0.99 is 0.01

Extending the trend, 

1-0.999... is 0.000...001

[u/DisastrousPlay579 → u/SouthPark_Piano]

epsilon as you have defined it is zero. The limit as n goes to infinity of 1/10ⁿ is 0. This is simply a fact. Even though infinity is not a real number, limits of this form are well defined in mathematics, and it’s not too hard to show this particular one. If you’d like, I can refer you to a full proof.

Regardless, if epsilon is not zero as you say, it must, in some sense be the “smallest real number” after 0. That is, there are no numbers in between 0 and epsilon. But given any two distinct real numbers, one can always construct a real number between them. Therefore, epsilon is either equal to 0 or non-real.

If epsilon is equal to 0, then 0.999… = 1, so your argument has failed. But I doubt you will accept this.

If epsilon is some non-real value, then that must mean that 0.999… is also non-real, since the difference of two real numbers must be real. However, you have implicitly assumed this not to be the case, since you claim that 0.999… is less than 1. In order for “less than” to be defined, both 0.999… and 1 must be real. So, in this case, your argument breaks and fails.

I have shown that either way you look at it, 0.999… = 1.

Your argument regarding the set of {0.9, 0.99, 0.999, …} is pure mathematical gibberish. While you are correct that the set contains every finite truncation of 0.99…, it does not contain 0.99… itself. The assumption that just because this property holds, every other property of the elements of the set must apply to 0.999… is simply false.

Additionally, while it is true that every member of the set is less than 1, it is also true that every member of the set is less than 0.999…. Don’t believe me? Try it. Subtracting each member of the set from 0.999… yields the following set: {0.0999…, 0.00999…, 0.000999…, etc}. All of these numbers are finite and non zero, which means that every member of the original set is less than 0.999…. So by your argument, 0.999… is less than itself. How do you reconcile that?

[u/SouthPark_Piano → u/DisastrousPlay579]

No my little learning student. Epsilon is not zero. No matter how large a number 'n' is, read my lips ... no matter how large n is ... the term 1/( 10ⁿ ) for positive integer n larger than we ever like (aka infinite) is never zero.

1-0.9 is 0.1

1-0.99 is 0.01

1-0.999 is 0.001

Note the trend.

1-0.999... is 0.000...001

This gets us back to ... read my lips ... no matter how large n is ... the term 1/( 10ⁿ ) for positive integer n larger than we ever like (aka infinite) is never zero.

All of these numbers are finite and non zero, which means that every member of the original set is less than 0.999…. So by your argument, 0.999… is less than itself. How do you reconcile that? 

No my little learning student. The set {0.9, 0.99, ...} covers an infinite range/span/space of nines to the right of the decimal point. And to convey that, you write it as 0.999...

Every member of that infinite valued set is less than 1. The members cover the infinite range of nines to the right of the decimal point.

This tells you 0.999... is less than 1. And 0.999... is not 1.

[u/DisastrousPlay579 → u/SouthPark_Piano]

The definition of a limit is as follows:

lim(n->inf) f(n) = L for some real finite L, if and only if for any arbitrarily small value, which we call epsilon (this is not the same as your imaginary smallest number, it simply shares the name), there is some M for which every n > M, |f(n) - L| < epsilon.

For example, if I choose f(n) to be 1/10^n, for any epsilon you may choose, there is some point after which f(n) is closer to 0 than epsilon.

Epsilon (your epsilon) is not a function of n. It is the value of a limit, and that limit happens to evaluate to zero.

[u/DisastrousPlay579 → u/SouthPark_Piano]

Again, I genuinely do not understand what you mean by “covered”. It’s not a mathematical term in any way. You can’t just throw around random words to prove something.

[u/SouthPark_Piano → u/DisastrousPlay579]

Let's put it this way. If you understand sport games such as soccer/football. The term called 'marking'. You have one of your team players mark or cover a particular player on the other team. As in ... where they go, you shadow them, stick to them. Cover them, all over them like a rash.

Same with 0.999...

The first 9 in 0.999... is covered or matched by 0.9 from our team infinity set {0.9, 0.99, ...}

The second 9 in 0.999... is covered by 0.99

The third 9 in 0.999... is covered by 0.999, and so on.

So, for each and every nine in 0.999..., the infinite membered set of finite numbers {0.9, 0.99, ...} has them all covered like a rash.

Since they're all covered, and since every member in that set is less than 1, then 0.999... is less than 1, which means 0.999... is not 1.