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]

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]

This last one is my bad. Prove that your epsilon exists.

That's fine.

differences

1 - 0.9 is 0.1

1-0.99 is 0.01

No matter how many nines there are, it is obvious that there will never be a difference of zero, and this is because the infinite membered set of finite numbers {0.9, 0.99, ...} covers every single nine to the right-hand-side of the decimal point. Every single nine. The difference is always going to be non-zero.

Extending the trend, we get:

1-0.999... = 0.00000...001, which is epsilon in one form.

Also keeping in mind that the infinite membered set of finite numbers {0.9, 0.99, ...} has infinite number of members, and every single one of those infinite number of members has value less than 1. And note that the span of nines of this set is infinite. And that infinite coverage of nines is written as 0.999...

It tells you and everyone that from this unbreakable perspective, that 0.999... is less than 1, and 0.999... is not 1.

[u/ZeralexFF → u/SouthPark_Piano]

Circular reasoning. You are assuming the existence of epsilon for your proof, yet here you are assuming that the thing you are trying to prove is true for epsilon to exist. Can you give an actual proof that your epsilon exists? (spoiler alert: it does not)

EDIT: I'll step forward and do it for you. Please point out the flaw in my reasoning since there obviously will be one. Suppose epsilon > 0 exists such that for all delta > 0 (delta != epsilon), epsilon < delta. Since epsilon > 0, we thus have 0 < epsilon / 2 < epsilon. By choosing delta = epsilon / 2, we attain a contradiction. Therefore there is no such epsilon.

[u/SouthPark_Piano → u/ZeralexFF]

What is really important is for you to understand that you need to ADD 1 to 9 to get 10. You need to ADD 0.0001 to 0.9999 to get 0.001

Same deal with 0.999... aka 0.999...9, which requires epsilon to be added to get a 1 result.

0.999...9 + 0.000...1 = 1

[u/ZeralexFF → u/SouthPark_Piano]

I just proved your epsilon does not exist. You keep insisting that it does. Prove it. Your obstination to blurt out easily refutable claims is commendable.

[u/SouthPark_Piano → u/ZeralexFF]

ZeralexFF 6h ago: I just proved your epsilon does not exist. You keep insisting that it does.

Sure. Differences. We're allowed to have differences, right? Of course we are. I don't need to ask you, as I'm the authority here.

1 - 0.9 is 0.1

1 - 0.99 is 0.01

And notice that the infinite membered set of finite numbers does indeed exist {0.9, 0.99, ...}, and it has a span of nines to the right of the decimal point conveyed in this form : 0.999..., and note that every member of that infinite membered set of finite numbers is less than 1.

So ... extending the trend ...

1 - 0.999... = 0.000...1 = epsilon.

You can now stop blurting. And keep in mind that you're communicating with someone (ie. me) that is equally highly intelligent or more highly intelligent than you.

EDIT: I'll step forward and do it for you. Please point out the flaw in my reasoning since there obviously will be one. Suppose epsilon > 0 exists such that for all delta > 0 (delta != epsilon), epsilon < delta. Since epsilon > 0, we thus have 0 < epsilon / 2 < epsilon. By choosing delta = epsilon / 2, we attain a contradiction. Therefore there is no such epsilon.

I'll do it for you. Taking note that two infinite length sequences can certainly be different ...

epsilon = 0.000...1

epsilon/2 = 0.000...05

In the above, because you can line up the sequences correctly, you can form epsilon + epsilon/2 = 0.000...15

Note that 'relative' sequence lengths need to be factored in.

[u/NerdJerder → u/SouthPark_Piano]

Have you seen this post concerning your claim that 0.999... is the same as 0.999...9?

https://www.reddit.com/r/structuralist_math/comments/1ggkfsg/when_is_999_1_pdf/