>>11041982>The proof that 0.9999999... is equal to 1 depends on assuming that 0.3333333... is equal to one third. Where's the proof of that?No it doesn't.
>Additionally, there's no proof that allowing infinite strings of digits followed by a finite string of digits is invalid. It's not invalid, it's just undefined. When you use new notation, you have to define what it means. One of the equivalent definitions of real numbers is that of cauchy sequences (which means sequences of rational numbers a_n such that for all rational x>0, there is a natural number N such that for all natural numbers n,m > N, |x_n - x_m|<x), and for an infinitely long decimal expression, mathematicians define the real number that it represents by taking a sequence of increasing finite truncations of the decimal expression. In particular, we write 0.9999... to denote the real number represented (read: which contains, as it is an equivalence class of sequences, the sequence) by the sequence (0, 0.9, 0.99, 0.999, and so on).
>People say that 0.3333333... ...3334 is not a number that can exist, but where is the proof?This is the same as saying "people say [d173g8 1c sss] is not a number that can exist, but where is the proof?". The problem is that you used new notation which doesn't have meaning in the standard literature. Hence you need to specify what you mean by that, otherwise you're just talking gibberish. 0.3333... and 0.999... both have standard interpretations, because real number system is a fundamental concept in basic mathematics every mathematician knows the meaning of a decimal expression. If you invent your own number system which can accommodate 0.999... != 1, then good on you, but you need to specify that you mean the other number system and then what you mean by those expressions in that number system, because OTHERWISE people will assume you mean real numbers as well as other standard definitions that go with them.
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