0.999999... = 1?

What the fuck. Why is that true. They got different numbers in them.

Ok, let's get serious here. 0.999... = 1. An "easy" way to see it, that does not involve any of the proofs shown above, is this:

Theorem: 0.999... = 1

Proof: Let R be the set of real numbers. R is know to be an Dedekind-complete ordered field. Since R is a field, addition and multiplication are defined with their usual properties; since a field is a group under addition, it follows that R is an ordered group, and therefore it defines an uniform structure. An uniform structure is complete, and therefore for every two members a,b in R, there exist infinite members (denoted by s) s such that a < s < b. Since there is no real number s such that 0.999... < s < 1, it follows that 0.999... = 1. QED

The idea behind the math: if two numbers are different, one can always find a number in between them. You cannot find a number in between 0.999... and 1. Therefore, 0.999... = 1

PS: (1 + 0.999...)/2 does NOT does the trick; if you evaluate the division correctly, you find 0.999... once again.