@Godcast My practice doing proofs is a bit hazy, but I know this one is no proof and quite a bit of bullshit.
Forget the 0.999... and the 1 and look at the lim_{n-> infinity}(1-1/n). By itself, what does that equal? Well, as n approaches infinity, you would get just 1 with that (correct me if I am wrong those who have done more proofs than I have).
You can't just throw in some numbers in an equation when you don't understand what they do and say it is a "proof." Leave your boyfriend, woman.
@Godcast Also, to my recollection, you cannot just factor out the limit part as if it is just a number in itself. Because it is not. So that is why the "proof" will not work, because he can't even use those things correctly.
Again, I have not been in practice using proofs, but I can tell this is all wrong when you try and say that two distinct numbers are equal when they are not.