No. I don't have a problem with:

0.9999... = lim[x -> infinity] (9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x)

And no. I don't have a problem with:

0 = lim[x -> infinity] (10^-x)

I have a problem with this:

It's your leap of faith. It's the conclusion you draw from a limit that I have a problem with; The fact that you treat the equality in the limit the same as you treat the equality in the conclusion of your proof. You don't bring the result of a limit into plain algebra without the equal sign changing meaning also. I've said this numerous, countless times. And here I am saying it again. I think that's enough of me saying this. Agree or disagree, but please do not again misinterpret my beef with your proof. That's bothersome.

Indeed. I think that makes an excellent wrap up. I have my own idea of why that is so; I defend the current axioms and mathematical language cannot handle these type of numbers and hence the inconsistency in proofs, and hence why new axioms and languages have been devised to treat these Real numbers in the context of infinitesimals and non-terminating decimals.

The current axioms however hold true enough for most mundane purposes. And they don't need to be revised. So, for almost all purposes 0.999... can be 1. I cannot dispute that. Unless there's a specific need for an infinitesimal to have a meaning other than 0 in real numbers (e.g.

1, or

2), why should we bother?

Again, thanks you all for the debate. Loved it. Cheers.