# 1 = 0.999…?

There is a proof of this that runs like so…

First let , the … (triple) dots mean an infinite number of digits of 9s.

Then multiplying both sides by , we get

Thus solving for , we subtract the first equation from the second getting

From this we obtain . This implies . So the proof goes.

My first question is, do people accept this as a valid proof? Apparently lots of mathematicians do. I find it hard to accept this proof. If anything, even if I accept the proof, I can only conclude that the system of real numbers does indeed result in a contradiction! The number can be subjected to equivocation. That is the problem. It is not obvious that what is meant here is that $0.999… = 3 (0.333…) = 3 (1/3)$. In which case there is not even need for a proof. It is a renaming exercise and no genius needed.

At any rate look at the proposed proof. I am skeptical in the technical sense due to the fact that we are not told what means. It gives the impression that one starts off with an assumption, , then one derives that . Philosophically as I understand real numbers, 0.999… is never the same as 1.0, the former always falls short from the latter. That is if it is taken as a value. In logical terms in the light of the proposed proof, this means one starts off with an assumption and then from it one arrives at . Hence , implying that . However, IMHO, this is a contradiction.

If anything, the proof system allows for a contradiction and that is not cool.

OK, not convinced of my objection? Would you then accept that ?