Noone said that.
The point is that both (0.\dot9) and (\infty) can be represented as infinite sums.
I am not.
That’s correct. (And yes, you are right, there are infinite sums that evaluate to a finite number. But that’s irrelevant.)
I take this to mean that you agree that infinity can be represented as an infinite sum. In other words, you agree that (\infty = 1 + 1 + 1 + \cdots).
That’s something that has to be proven. One way to prove it is by doing arithmetic with infinite sums. If you can’t do arithmetic with (1 + 1 + 1 + \cdots), because it equals to (\infty) and because (\infty) is not a quantity but a quality, what makes you think you can do arithmetic with (0.9 + 0.09 + 0.009 + \cdots)?
But (\infty) can be represented as an infinite number of terms.
Are you telling me you don’t really know how to do arithmetic with infinite sums?
They had to. One of their proofs is based on the assumption that (0.\dot9 - 0.\dot9 = 0). That’s the same kind of assumption that leads to (1 = 0).
(\infty + 1 = \infty) // subtract (\infty) from both sides
(\infty + 1 - \infty\ = \infty - \infty) // substitute (\infty - \infty) with (0)
(1 = 0)
What’s wrong here is that due to the meaning of the symbol (\infty), it does not necessarily follow that (\infty - \infty) equals (0). It’s indeterminate.
This proof does exactly that. It assumes that the difference between the two underlined infinite sums is equal to zero. Hence the wrong conclusion.
Fine. In that case, you have to accept the conclusion that (1 = 0). But you don’t because you don’t like it – I don’t like it either. We already know that (1 \neq 0), so this is an indication that the proof is fallacious. But where’s the mistake?
Well, in that case, you disagree with this Wikipedia proof because it does arithemtic with infinite sums.
The only way out, it appears to me, is to claim that you can do arithmetic with SOME infinite sums but not ALL of them e.g. you can say you can only do arithmetic with convergent sums (such as (0.\dot9)). But in that case, you’ll have to explain why. And you can’t say it’s because you don’t like the consequences of doing arithmetic with divergent series. That’s not an acceptable answer.