It is NOT a formal proof! The reason being that you're assuming that you know what it means to subtract infinite decimal expansions from each other. Mathematicians tend not to be very happy when you invoke infinity, because it can be very counterintuitive.
Please let me preface this response with an acknowledgement that I know we are both engaging in quite a derail from the OP, but unless someone tells us to stop, I don't see anything wrong with it, as long as we are willing to continue the discussion.
OK, onto the discussion.
First,
no one is particularly comfortable discussing infinity. You're right; it's highly counterintuitive and gives everyone fits. It's a terrifically handy concept, however, so we mathematicians and our academic cousins, those delightful oddballs who call themselves "physicists," cannot get around it. At some point, we simply have to dig in our teeth and choke down the concept, then attempt to digest it a bit (see, I really am a mathematician rather than a verbally talented person; I'm not very good with metaphors).
Second, you don't really have to discuss infinity too deeply to grasp on a tangible level the concept of a repeating digit after the decimal that goes on forever. Seventh grade math students tend to accept and grasp it on a level not much higher than a visceral one.
Third, you don't have to assume that "you know what it means to subtract infinite decimal expansions from each other," as you say. Any 7th grade math student is supposed to understand that if you subtract 3 from 3 (or X from X), you get 0. Once a student grasps that, and you have explained what it means for a decimal to repeat indefinitely ("forever" might be the most effective way to describe it in secondary school), then it follows easily that subtracting .999... from .999... equals 0. That, multiplication by 10, and dividing 9 by 9 are the only concepts a student needs in order to balance each side of the equation.
You want the proof to be more formal? Jeez, do I have to state my axioms for you? Do we really do that in the 7th grade, or are they more implicitly understood? To the best of my recollection, we don't have to state them in our proof in the 7th grade. I don't recall hearing the term "axiom" in a mathematics class until college calculus (I must have slept through it in AP high school calculus). Of course, from then on, it was a regular part of every proof.
I think it's possible that you may be burdened in a sense by your mathematics education in approaching these problems, and in particular in your approach to my proof. I get why or how someone might be "too educated" in a sense to relate to it on a simpler level. Nevertheless, I recall how I responded to this proof when I first saw it in the 7th grade, and it was not hard to grasp then. I had never been exposed to the more advanced concepts of infinitely repeating series (I know, I know, that's what .999... is, but at the 7th grade level a student can accept "repeats forever" as a shorthand definition and understand it) converging series, or limits. We didn't go into a lengthy discussion or explanation of what "infinite" means either. I think it's entirely unnecessary at the 7th grade level.
Take 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... . (Ok, first I should prove that this sequence converges, but I'm too lazy to do it right now
) It has a certain value. But, by rearranging the terms, I can make this sequence converge to whatever the heck I like. (see end of article "Alternating series" on Wikipedia.) And it's knowing about infinity-related loopholes like this that would make a mathematician lose sleep at night having seen your first "proof". You don't know what precisely this 0.999... object is.
So, for that reason, you have to clarify exactly what you mean by 0.999..., without using infinity, to formalise the proof.
Please see my burdened by education comments above. Understand please that I do not mean them to be insulting or denigrating.
My proof is sound, solid, and formal enough for the appropriate and intended audience. I agree with Dr.Kitten that if you want to dissect the hell out of it in a post-graduate seminar, you can spend a great deal of time examining and defining the concept of 0, for instance. That could be the subject of an entire semester's worth of study, and it might prove very interesting. You simply don't need that level of analysis in order to apply it in the setting of my proof.
AS
