Your proof equalizes an infinitesimal to 0 based on the established rule that, there not being a distance between any two numbers, those two numbers are the same. That's the genesis of the proof. Sang-drax proof just above me uses that rule to prove that 0.999... equals 1 without that requiring establishing the identity of an infinitesimal. Essentially you both prove the same thing though, by slightly different routes.

However I find this argument curious (emphasis mine):

But isn't that also an argument that contradicts the whole thing? I can reword the last sentence to an equivalent sentence that puts things in a different perspective: "There is always aOriginally Posted byMacNillyrealnumber between 0.999... and 1 (no matter how many 9's you choose, I choose one more 9)."

This is at the core of the concept of infinity. Where you wish to establish you know what happens when we reach infinity, I wish to establish we actually don't. Where you wish to establish you can prove it, I wish to establish you can't. And my own proof for that theory is exactly the contradiction that the proofs we have seen so far present when we consider that for any real number we can't think of, 0.999... is never equal to 1.

This is the basis of my thesis that you can't prove 0.999... equals 1. You can only establish it through an unproven rule that serves the purposes of limits and that gives consistency to the Real number system (that unproven rule being that there is no distance between 0.999... and 1).

I can certainly accept the ruling. But cannot accept that as a proof.