In your proof you say >And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infini

In your proof you say >And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's. This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.