Of course I'm serious. And the only reason there are 25 pages is because it isn't a fact, and no one has been able to show otherwise!

Well, your question is too ambiguous. You haven't said what type of numbers 0.999.. and 1 are. The square root of infinity might make no sense to someone who works with real numbers, but it's perfectly well-defined if you're working with surreals.

IF you cobbled together a definition based on decimal expansions, for example based as a sequence (a,b1,b2,b3,...) of integers, and worked from there, sure, you could get to where (1,0,0,0,..) != (0,9,9,9,...) But you wouldn't have many of the useful properties of the reals, such as bijective correspondence to the number line. Almost every undergrad analysis student thinks of this as an alternative to the cumbersome buildup of the reals from Peano axioms through Dedekind cuts or Cauchy sequences.