> You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
I _know_ that.
I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault.
I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly.
Feel free to disagree...