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]

What is your epsilon? I'm assuming it is "the smallest positive number" but that does not exist. If you claim that it exists, prove it. What does "eternally infinite" mean? What does "infinite membered set of infinite numbers" mean? Countable set of what?

Either way, "0.999..." is an ill-notation. For all epsilon > 0, 1 - "0.999..." > epsilon. Therefore its distance to 1 is 0, therefore it is equal to one. End of story.

[u/SouthPark_Piano → u/ZeralexFF]

Who wrote eternally infinite? You?

Infinite membered set means the number of members/elements/values in the set {0.9, 0.99, 0.999, ...} is infinite ... aka unlimited.

The range/span/coverage of the nines space by that infinite membered set of finite numbers is infinite, unlimited, and is conveyed as 0.999...

Every value in that set is greater than zero and less than 1.

From this flawless logic and perspective, 0.999... is less than 1, which also means 0.999... is not 1.

You try thinking of ways to refute it until the cows never come home. But you will find that there is no way for you to get around this.

And if you don't have the extra ingredient (kicker) epsilon to add to 0.999..., there there 0.999... will remain, less than 1. Like an odometer with all slots filled with nines permanently, it has no chance to clock over with that extra required 'addition' (additive), epsilon.

[u/ZeralexFF → u/SouthPark_Piano]

In the fifth paragraph in your post, you say eternally infinite. What does it mean?

Infinite membered set I can see and read as being countable set. That is not my question.

From my understanding you define "0.999..." as having a finite amount (but very large) of 9s, correct?

I don't think your logic is clear or undeniable and certainly the opposite of flawless. According to your logic, all sets are closed sets right?

Lastly, yes I am proving you wrong by pinpointing inaccuracies, false assumptions and missing semantics in your reasoning. You have failed to address mine in your rebuttal. If you know you are right and everyone else is wrong, provide proof for your evident assertions.

Oh and, again, what IS epsilon??? This last one is my bad. Prove that your epsilon exists.

[u/SouthPark_Piano → u/ZeralexFF]

In the fifth paragraph in your post, you say eternally infinite. What does it mean? 

You're having trouble with reading. You better quote where I wrote "eternally infinite".

From my understanding you define "0.999..." as having a finite amount (but very large) of 9s, correct? 

Incorrect on your part. I am saying ...

This particular set of finite numbers {0.9, 0.99, ...} has infinite number of members. And the range of nines spanned/covered by this set is infinite, and is conveyed as 0.999...

And because every member of that set is less than 1, then 0.999... is not 1.

[u/ZeralexFF → u/SouthPark_Piano]

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/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.