>>11594817As a onetard myself (I firmly hold that 0.99...=1) I agree with OP that the "proofs" many people provide in these threads are not proofs at all and circular. I commend OP for recognizing faulty logic. There are mathematically illiterate people on both sides.
The truth, OP, is in how you define 0.9999...
If you want all infinite decimal expansions to represent actual numbers the rationals don't suffice and you have to deal with equivalence classes of Cauchy sequences or Dedekind cuts.
Although if you're willing to accept that not all decimal expansions have meaning (so you can stay in the rationals), then using the definition that:
for a sequence of digits a_1, a_2, a_3.... (formally defined as a function f: N -> N from the naturals to the naturals where f(n)=a_n) the number 0.a_1a_2a_3..... is such a number x (if it exists!) that for all rational numbers (i.e. fractions) e>0, there exists a natural number N such that for all n>N,
|0.a_1a_2....a_n - x| < e.
With this definition, we can take each a_i to be 9. Then all that 0.999...=1 is claiming is that for all rational numbers e>0, once you add enough digits, the finite decimal expansion is as close to 1 as e is to zero.
It's rather simple and clearly true. And remember, all it boils down to is how you define 0.999...
How do YOU define 0.999..., OP?
