>>10
That's horrible maths.
Basically, one way to construct the real numbers from the set of rationals, which are easily constructed from the naturals, which can be constructed from the empty set, Is to define each real number as the limit of a cauchy sequence.
From this definition and the basic definition of a limit it's obvious that 0.9999~ = 1 as 0.9999~ is defined as the limit of the caucy sequence 0, 0.9, 0.99, 0.999, 0.9999, 0.999999 etc..
And the limit of that sequence is trivially one.
Proofs based on manipulating fractions are quite useless, because the properly use them we have to show that 1/3 = 0.333333333~ or assume it as an axiom, but that's the basic property that we're trying to prove.
>>3 is also a flawed proof, before you can manipulate recurring decimals like that you first have to prove that you can treat them as normal numbers, which would involve defining them as a limit and proving simple limit laws like lim a + lim b = Lim a+b etc.