0.999... = 1 (?)

Sorry for reviving this thread, but I'm baffled by this. My Calculus teacher, a Ph.D. in Mathematics, told me today that the closed set [0,1) has no upper bound. Specifically, I asked him the question, "what is the highest possible number that can fit in that set"? He replied,"there is no highest number". Then I proposed that the highest number might be 0.999...9! and he replied that it isn't, because it equals 1, and that trumps the definition of the closed set.

=[

It's not too hard to understand, really.
Assume that you have a real number x such that 0 <= x < 1
If you furthermore assume that this number is the highest number existing in this set, then the equation
x < 1 has to be satisfied, and x + ε < 1 implies that ε = 0. Now, choose ε = (1 - x) /2, which implies that x + 2ε = 1.
Then, x + ε < 1, but ε > 0 because x < 1 => 0 < 1 - x. This results into reductio ad absurdum, which means some of our previous assumptions were wrong. As we can find a number 0 <= x < 1, then that means this x is not the highest number in this set, and that it is true for all x.

Noah

Can we add one vote to yes for Maplestory since you never see 99.99% in your experience bar?

Can we add one vote to yes for Maplestory since you never see 99.99% in your experience bar?

Not quote the right argument