/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

1. You haven't properly defined what the symbol 0.99... means, and thus cannot assume that algebraic manipulations (such as addition, multiplication) can be applied to it in any meaningful way.

2. You've defined it as a limit, that's at least a start, however you've applied a theorem on finite geometric series, as a limiting case, to this series.
This works, but you haven't proved that it does, so this one isn't a proof either.

3. As I said last time, this is the closest to a proof.
All that's lacking is you need to state what you're saying in a better way.
The real numbers are a hausdorff space, so given any two distinct points there are two disjoint open sets containing them. However for any open set containing 0.9999... it must have 0.9999.. + e for some e>0, but 0.9999...+ e > 1 for any e >0. Therefore you can't find two such sets, therefore 0.999.... and 1 aren't distinct points.