/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

>>74

1. Well you have to define somehow, otherwise it's just a symbol, it has to have some sort of mathematical interpretation.
Generally you define an infinite decimal expansion 0.a_0a_1a_2..... to equal the limit of the infinite sum a_i*10^-i.

It's a pretty easy job to show this converges for all value of a_i. during this you'll prove a proof of the form of 2. ie that all numbers which have an infinite string of 9's in their decimal expansion have two equivalent decimal expansions.

The reason your proof of 2 doesn't work is because you're applied a result that is derived to work on finite series, and taken a limiting case.

For example apply your formula to the series 1 + 2 + 4 + 8....
you get the answer -1, nonsense (in a sense anyway). You have to prove that it still holds for any common ratio < 1.

Now once you've prove all this you can start proving that addition and multiplication work for decimal expansions in the way you imagine (ie. that the new series you get still converges and converges to the value you want) and is well defined (ie, if you have A=A', B=B' for equivalent decimal expansions then c*A = c*A' and A+B = A'+B')

Then you can apply your second proof.

tl;dr? Maths is harder then you think.