Wikipedia proofs treat these symbols as representing infinite sums.
(0.\dot9 = 0.9 + 0.09 + 0.009 + \cdots)
Remember that (\infty) has an infinite sum equal to it which is (1 + 1 + 1 + \cdots). This means that if you can’t do arithmetic with infinities (because they are qualities, as you say, and not quantities) then you can’t do arithmetic with infinite sums either. (Which would invalidate Wikipedia proofs.)
Unless, for some strange reason, you don’t think that infinity can be represented as an infinite sum. In that case, I could simply stop talking about infinities and start talking about. . . infinite sums. There would be no difference with regard to my argument.
Do you agree that infinite sums come in different sizes?
Do you agree that ( (1 + 1 + 1 + \cdots) + 1 > 1 + 1 + 1 + \cdots)?
If you do, thank you very much.
But if you don’t, this means that:
((1 + 1 + 1 + \cdots) + 1 = 1 + 1 + 1 + \cdots)
Do you agree that (1 + 1 + 1 + \cdots = 1 + 1 + 1 + \cdots)?
Remember that one of the Wikipedia proofs claims that (0.9 + 0.09 + 0.009 + \cdots = 0.9 + 0.09 + 0.009 + \cdots).
If you don’t agree with this, you also don’t agree with Wikipedia proofs.
If you do agree, let’s subtract (1 + 1 + 1 + \cdots) from the above equation.
What do we get?
We get (1 = 0).
Do you agree with the conclusion?