/prog/rammer here, need urgent help

hello /sci/, long time no see
There is currently a troll on /prog/, and would greatly appreciate your help.

Basically, the troll is trying to prove that 0.999~ =/= 1
Here are some of his arguments
http://stormtower.invisionplus.net/?mforum=stormtower&showtopic=97

I'm wondering if the maths genius' on this board can rip him apart? Thank you

Maybe your third proof attempt can be turned into a proof.

Informal proof sketch/idea for #3.

1. Fact: a real number is uniquely described by a finite sequence of digits, followed by a decimal separator, followed by an infinite sequence of digits
2. Fact: The amount of real numbers is uncountably infinite.
3. We can define the successor function s : R -> R for real numbers as follows: Let "x.y" be a representation of a real number, where x is a finite sequence of digits, and y is an infinite sequence of digits. Then s(x.y) = x'.y', and x'.y' are obtained by incrementing the last digit of y.
(It should be provable that from this definition, for all reals x.y, x.y <= s(x.y); and also any real is s(s(s(...(0.0...)))...), i.e., some above-infinite amount of applications of the successor function)
2. If 0.999... != 1, then by (2) there should be an uncountably infinite amount of numbers between 0.999 and 1.
3. But s(0.999...) = 1.
4. (2) and (3) form a contradiction. Hence 0.999... must be 1.

Something like that :D