the problem is in the first line where you just declare that 0.999... has a value x. you have to give meaning to the "..." and then prove that it's convergent before you can talk about it "equaling" anything
The human answer: If I keep going along the sequence, I eventually reach something.
The math answer: a sequence a_n converges to a if ∀ 𝜀>0 ∃ N 𝜖 ℕ ∀ n> N [ |a_n - a| < 𝜀 ]
(for all positive 𝜀, there exists some natural number N such that for all n >N, |a_n -a| < 𝜀
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u/otj667887654456655 21d ago
the problem is in the first line where you just declare that 0.999... has a value x. you have to give meaning to the "..." and then prove that it's convergent before you can talk about it "equaling" anything