chaja...@mail.com wrote: > I have written a proof that 0.999... cannot be equal to one in the > system of real numbers. . > While at the end of it all you may not fully agree with my proof, much > I as have never seen a proof asserting they were equal that I was able > to consider valid, . That's the problem right there.
You see, there *are* valid proofs that .999... exactly equals 1 in the system of real numbers (they might be unequal in an enlarged system, that includes infinitesimals) and therefore the people who have seen and understood those proofs already know that your proof, even if it is completely original, still can't possibly be right.
Here's one of those proofs:
The difference between 0.999 and 1 is 0.001.
The difference between 0.9999 and 1 is 0.0001.
In the system of real numbers, two numbers are not equal only if some number that is greater than zero is the difference between them.
For any number greater than zero, there is some number of the form 0.000...001 that is less than that number.
So any number that is close to 1, but is slightly less than it, is less than 0.999...999 where the string of 9s is finite in length.
Thus, 0.999..., because it goes on forever with nines after any of the strings of nines that stop, can't differ from 1 by any finite amount. Therefore, it is equal to 1 in the way that equality is defined for the real number line.