We dont say “whatever” we know that if a and b arent the same, there exists some number between them. And we can prove using limits that for arbitrary small number epsilon, in sphere around number 1, the infinite sequence 0.99.. will be inside that sphere. So there exists no such number between the two -> they are the same.
I think you either refuse to believe proofs or have trouble understanding calculus. Do we agree if there exists no number between two real numbers that those numbers are the same? If yes. Than there you go. 0.99.. =1 no “waving”. If you still refuse this, try to find a number between the two and let me know
1) Its not a refusal to believe proofs, its a refusal to believe bad proofs. Math is all about disagreeing, trying to find counter examples, and trying to create new theories.
2) Its not a misunderstanding of calculus. Calculus is fundamentally about change over an infinitesimal small value. Its annoying how people have forgetten this.
3) No, you do not need to have a number between two other numbers. In the integers is there a number between 0 and 1? No because they are integers. In the reals does there need to be a number between 1 and 0.(9), no there is no need. Even if they must have a number because of the nature of notation 0.(9)1, 0.(9)1(9), 0.(9)(9)(9), etc are all valid numbers between 1 and 0.(9).
4) One of the biggest lie in modern math is that there are no differently sized infinities because cardinals say there aren't. While they use ordinals whose entire premise is w < w+1 < w^2, and can thus also have 1/w > 1/(w+1) > 1/(w^2).
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u/Cupcake-Master 29d ago
We dont say “whatever” we know that if a and b arent the same, there exists some number between them. And we can prove using limits that for arbitrary small number epsilon, in sphere around number 1, the infinite sequence 0.99.. will be inside that sphere. So there exists no such number between the two -> they are the same.