> Using the standard definition of the real numbers together with the > standard definition of an infinite decimal as a limit, it's easy to > prove that .999... _is_ equal to 1, hence it's automatic that your > proof is flawed.
Taken on those terms I would have to agree with you within the confines that the limit is only allowed to yield real numbers.
But that's fine, preclude out of hand that my proof can have no validity without even reading it.
I cannot accept that 0.999... is literal equal to 1 when we teach this to students without the concept of limit. Everyday on message boards across the world, some new student mentions that they learned something that seems wrong, that 0.999... is exactly equal to one. No one ever told them the limit as the places grow is 1. This is also pretty obvious to everyone and the student would have had no objections.
My claim is not a claim in terms of limits. I have defined my number as an infinite decimal and not mentioned limits. This reflects how I and most student across the country were taught this issue. I later mention limits in my analysis/proof only to say that limits do not reflect the entity that yields them. My proof is standing up for that part of our understanding that made every single one of us feel that 0.999... equally one seemed off somehow.
No you defend the ideas wrapped up in new ideas that enforce and allow it.
I still saw that take the "inferred" limit away from this infinity decimal, and in the system of real numbers it is strictly less than 1.