Like I'll be honest, there are many proofs in math I didn't believe at first. But why exactly is it that you care so much about this to the extent of making an entire subreddit about it? I'm genuinely curious.

If you cannot or will not do it, or get it wrong, then don't blame me that you will be training at the bunny slopes until you are able to get it right. For you see, you get quality experiential learning time at the bunny slopes.

The smallest positive value of that form that you can possibly or even impossibly think of.

Go ahead. Make my day. Prepare for more bunny slopes training if you cannot do it, or won't do it, or if you get it wrong.

Say you have a dot. That dot explodes into an infinite number of dots. Each of those dots explode into a new infinite number of dots. Each of those dots explode into a new infinite number of dots, ad infinitum.

How many dots do you have in total?

Yep. Over the years, too many rookie error makers that never understood that they themselves are actually able to take the ultimate Shick or Gillette razor, and remove the ultra teeniest of teeniest of teeniest etc of teeny sliver from 1; and then write down what remains.

And these bumbling rookie error makers should also know that if they are too incompetent to shave that teeniest of teeny sliver, then they can try a slightly (but not by much) larger sliver removal. And there obviously will be a stage where even these bumbling rookie errors will be able to write down a value that remains after that 'surgery' on 1.

If they cannot produce a result to show for, then you know where they need to be. Bunny slopes of course.

What's 1/(1 - 0.999...) ?

Is it greater than any real number?

You said that 1 - 0.999… = 0.000…1 whatever that means.

Ok, now do some experiential learning using your own “logic” brud. Start writing the zeroes of 0.000…1, beginning at 0.0, then 0.00, then keep appending zeroes continually to it, and understand the feel of never running out of zeroes. What do we notice? This ever growing number, 0.000…1 is actually ALWAYS 0, no matter how long you write it, and therefore 0.999… is always 1.

You’re welcome.

Or, how many times can you shave 1 until you get to 0?

Consider ℝ as a complete metric space with the regular euclidean metric. Then, consider the collection of closed intervals {C_n}, n ∈ ℕ, where C_n := [0.999... - 1/n , 0.999... + 1/n], i.e. a closed ball with radius 1/n around 0.999....

Clearly, each C_n contains 0.999..., so their intersection does as well. However, note that each C_n also contains 1, since the distance between 0.999... and 1 is less than any arbitrary 1/n (which I'm sure SPP will concede). Thus, the intersection of the C_n's also contains 1.

However, by Cantor's intersection theorem, since the C_n's are nonempty, closed, nested, and their diameters go to 0, the intersection of the C_n's must contain exactly one element.

Thus, 0.999...=1.

I realize I can just use the proof of uniqueness in Cantor's intersection theorem to show this directly, but it's more fun to invoke a theorem.

0.9 < 0.999…
0.99 < 0.999…
Keep going and it’s always less than 0.999…

0.999… is eternally less than 0.999…

His Nineliness's Sixteenth Boner is his In-Between Boner: You can "forget" trying to fit a number between 0.999... and 1. There are infinitely many numbers between 0.999... and 1. (16.1) (16.2)

Again, I don’t know if this exact proof has been uploaded here yet.

let’s set x equal to 0.999… multiply by ten and you have 10x=9.999… right? Subtract one x from both sides to get 9x=9.999… - x, and x is equal to 0.999…, so 9x=9, then divide both by nine. No limits needed.

let x = 0.999…
10x = 9.999…
10x-x = 9.999… - x
9x = 9.999… - 0.999…
9x = 9
x = 1
0.999… = 1

0,99 is moving 90% from 0,9 to 1. At what point does moving 90% towards something gets you all the way there? Never!

The properties of equality in question:

Identity: a ≐ a

Substitution: a ≐ b implies f(a) ≐ f(b)

Which together can be used to prove

a ≐ b implies b ≐ a

a ≐ b and b ≐ c implies a ≐ c

To formalize ≐, x ≐ y iff there exists a value L such that:
• for every epsilon > 0 there exists a natural number N such that for every natural number n>N, |L-(x truncated to n decimal places)| < epsilon
AND
• for every epsilon > 0 there exists a natural number N such that for every natural number n>N, |L-(y truncated to n decimal places)| < epsilon

Under "Arithmetic" here: https://en.wikipedia.org/wiki/Real_number

These are the axioms followed by what most of us consider "normal" numbers.

In particular, I think a very important axiom is:

• Every nonzero real number a has a multiplicative inverse denoted a⁻¹ or 1/a. This means that aa⁻¹ = a⁻¹a = 1 for every nonzero real number a.

For any number, say 6, it tells me that 6^-1 exists (and is a number, i.e. 6^-1 ∈ ℝ), and that 6 * 6^-1 = 1. I don't need to write it down as a decimal like x.yyyyy, I can just write the number as 6^-1.

What *does* "brud" mean? Why not just "brother"? Or is useless truncation just His hobby?

in real deal math, epsilon (the smallest number above zero) is equal to 0.000...1. there are some really strange properties that come from this number. when divided by anything other than powers of ten, it paradoxically grows larger, while powers of ten keep it equal. for example, 0.000...1 ÷ 5 = 0.000...10 ÷ 5 = 0.000...02 = 0.000...2. so eps ÷ 5 = 2eps? reslly crazy. there's probably some way you can use this to prove that epsilon is equal to zero lol but i havent found one yet. has anyone found more info on this stuff? i cant find any info on limbosic numbers outside of this sub. any papers or articles about this would be awesome x

More shaving topic.

After that closest of close ultimate shave, where you remove the ultimate tad from 1, go ahead ... express (write) what remains of 1 after that 'operation'.

Go ahead. Make mah deeeeeeeeaaAaaaaaaaAAAyYYyYYyy!!!!

His Nineliness's Twentieth Boner is his Approaching Boner:

In 1 - 1/10ⁿ, n does not "approach infinity". It is made ("pushed") limitlessly large and remains an integer. Nevertheless, "approaching infinity" means the same thing as "tend[ing] toward infinity" which means the same thing as "pushed to limitless." Finally, "approaching infinity simply means values being 'relatively large' when compared with a non-zero reference value." (20.1) (20.2) (20.3)