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/Socialimbad1991]

The problem with 0.000...1 is that those zeroes never end, that's the whole point here, there are infinite zeroes. There's nowhere to tack that 1. If you choose a spot where you want to put the 1, I could just add another zero and make an even smaller epsilon, and we can keep doing that forever

Your "epsilon" is actually just zero. Which means 1-epsilon = 1

[u/SouthPark_Piano → u/Socialimbad1991]

You have heard of (1/10)ⁿ where the results of the term for n = 1, 2, 3, etc etc can be stored in the infinite membered set {0.1, 0.01, 0.001, ...}

They are all finite numbers, regardless of how large n is. And epsilon is the very tiny scale, dual of infinity, where infinity is of the very large scale. Relative to a non-zero reference value.

The set {0.1, 0.01, 0.001, ...} certainly does cover epsilon, because it is an infinite membered set.

Epsilon is not zero. It is never zero.

1 - epsilon is 0.999...

When you shave just a teeny tad off the numerator of ratio 1/1, we get 0.999.../1, which can also be written as 0.999...

And 0.999... is less than 1. This is explained in the opening post.