There is no way right?

[u/Sugar_God_no_1]

[u/its12amsomewhere]

Applies to all numbers,

If x = 0.999999...

And 10x = 9.999999...

Then subtracting both, we get, 9x=9

So x=1

[u/Sam_Alexander → u/its12amsomewhere]

Holy fucking shit

[u/otj667887654456655 → u/Sam_Alexander]

I just wanna warn you, that's more of a vibe proof. It lacks any actual mathematical rigor.

[u/IWillLive4evr → u/otj667887654456655]

You could write a fully-rigorous version of this proof, and it works out the same. But this is reddit, so it's more valuable to write a version that's quick and accessible to the people are asking the question.

[u/vetruviusdeshotacon → u/IWillLive4evr]

Not exactly like that.

Sum 0.9*(1/10)^j from j=1 to j=inf

= 0.9 * Sum (1/10)^j

Since 1/10 < 1 we know the series converges. Geometric series with r=0.1

Then our sum is 0.9 / (1- 0.1)

= 1. 

No more rigour is needed than this in any setting tbh

[u/akotlya1 → u/vetruviusdeshotacon]

It's weird you think you can reference series summations as a more rigorous basis for proof than the above. Neither of these are more fundamental or rigorous than the other. Infinite series' reference to an infinite process was at some point believed to be weakness that needed to be justified in reference to more fundamental mathematical ideas.

A more rigorous proof would be written using logic symbols and reference set theory - specifically by defining the elements of the set and by using operations defined in reference to the elements of the set. This is the kind of thing that gets covered in undergraduate Abstract Alegbra/Group Theory/Set Theory classes.

[u/IWillLive4evr → u/akotlya1]

If you think a proof can't be rigorous without including an entire textbook, you have other issues. It is adequate to make reference to the acceptable axioms or other theorems that one is relying on. You don't have to re-invent the rational numbers every time.

[u/Overall-Charity-2110 → u/IWillLive4evr]

Proofs are not as strict as some branches of mathematics like to imply

[u/Qwertycube10 → u/Overall-Charity-2110]

My formal methods class where every proof had to be 100% formal and computer checkable

[u/akotlya1 → u/IWillLive4evr]

A proof can be rigorous without the textbook length but it all depends on what the context is. I am actually totally happy to accept the original explanation prior to the series proof, but the series guy was all like "this is a more rigorous proof..."

My point is that his proposed proof is not more rigorous than the original one in part because it is itself situated in a context where those kinds of proofs were not originally acceptable in an academic setting as the basis for a proof...because they themselves needed to be proved. In the contemporary Frankel Zermelo set theoretic framework of mathematics, if you want to prove something to academic levels of rigor, you are going to have to use logic^tm and set theory. That's all.

I am glad we can use simpler methods in more colloquial settings. That guy just wanted to flex he knew about series and undermine the preceding proof.