/sci/ - Science & Math

lim n -> infinity 2^n - 1 > 0 edition

Now that the dust has settled...
Who was right? Has the debate resulted in a conclusive answer?

https://youtu.be/x-fUDqXlmHM
We put an end this debate today. Both viewpoints on this topic are valid, but what we will be discussing is whether the viewpoint of 0.999...=1 is more or less reasonable than the viewpoint that 0.999... has no meaningful answer.
In the video above, it is proven that, if you choose to believe they have any meaningful answer, ...999 (or 999... idk) is equal to -1 and ...999.999...=0.
If you accept that 0.999...=1, then you must also accept those additional two statements as correct as well otherwise your logic is incoherent.
My question then is, how is it possible that a POSITIVE integer that extends infinitely to the right of 0 on the number line and therefore holds true that x>0, can be equal to the NEGATIVE integer -1? The same question applies to ...999.999..., in that how can a number with VALUE can possibly be equal to 0? I find it a bit absurd to think that these statements have meaning, i mean its not like ...999 is so infinitely large that it goes all the way around the number line and comes back around to -1.
I argue that, despite being a valid stance and absolutely being correct if it did have any meaning, 0.999...=1 is similarly absurd due to following the same logic, and it is a more reasonable stance to think that all three of these statements do not arrive at a meaningful conclusion.

Show me where the number 1 is in this image and I will concede.

as you all know, the math undergrad cultists have for some time told us that 0.999=1. Effectively the 0.999=1 meme is claiming that the sequence of partial sums 0.9,0.99,0.999... is convergent. Since a real valued sequence is convergent iff it is cauchy (Barnett 2015), I shall disprove that this sequence is cauchy to put this matter to rest. Let xn=0.99999... be a sequential term with n 9s. Let N be an arbitrary index. We take epsilon to be 0.000...001>0. now consider any indices j,k >= N. Then,

|xj-xk|=|0.999...(j times)-0.999...(k times)|
=0.000(min(j,k) times)...999...(|j-k| times)
>0.000...(max(j,k) times)...1
>epsilon

Hence 0.999... is not cauchy. I do not know why so called mathematicians still spew this propaganda. Personally I think it's an element of societal control, but feel free to discuss your own theories.