/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

How's this:

0.999.... = (0.9 * 10^0) + (0.9* 10^-1) + (0.9 * 10^-2) + .....

Infinite geometric series, etc.

0.999.... = 0.9 / (1 - 10^-1)
0.999.... = 0.9 / 0.9
0.999.... = 1

I'm sure I'm missing a few rules/theorems/etc....or all of them. Just tell me if I'm right here.