Noble has the right idea.
0.9999 recurring forever equals 0.9999 recurring forever and 1 equals 1. 0.9999 recurring forever "approaches one and for simplicity can be represented as 1".
The definition of limit does not claim that a function is equal to it limit. It merely states that the difference between the function and its limit can be shown to be smaller than any finite value.

You must be careful about setting a function equal to its limit. For example.
  • Limit[ (1 + 1 / n)^n ] approaches e (2.71828...) as n grows without bound.
  • Limit[ 1 + 1 / n ] approaches one.
You cannot say that 1^n = e for infinitely large n, which is what you get if you set functions to their limits here and try to specify what is happening for infinite values. Mathematicians avoid statements about what happens at infinity.

Fried Egg: I have never heard that a proof must be in some sense reversible in order to be valid. Is there a text somewhere that makes this claim? How do you define the reverse of a proof? Such a definition would have to be on a case by case basis. Certainly a single definition would not cover all the possible methods of proof. It seem particularly difficult to define the reverse of a Reducto Ad Absurdum proof. What is the reverse of Wiles proof of Fermat's Last Theorem? What about disproving a statement by showing a counter example? Can such a disproof be reversed? I do not remember any book or text that showed a proof, and then went on to validate it by showing it in reverse.

I can prove that a particular number is not a prime by showing that it is the product of two particular numbers. 7 * 13 = 91, therefore 91 is not prime. What does it mean to reverse this proof? Can you determine the factors from the conclusion that 91 is not prime? The forward proof is simple, how do you do it backwards without using information from the forward proof? 91 / 7 = 13 is the only approach I can think of and that uses data from the forward proof. Godel proved that certain axiomatic systems are either inconsistent or incomplete. What would be the reverse of such a proof?

Most ordinary geometric proofs seem reversible. Given the Pythagorean relationship, I suppose you could prove the triangle to be a right triangle. Many other proofs are probably reversible. After all, if a forward going proof seems valid, you would hardly expect a reverse proof to be invalid. It seems reasonable to assume that most valid proofs are in some sense reversible. It does not seem reasonable or necessary to require reversibility for validity.

Others: The proof that recurring .9999 equals one seems reasonable, but does not seem valid to me in spite of what some calculus text might say. I do not question its validity due to lack of reversibility.

I think the proof is invalid because of the multiplication of recurring .999 by 10 and subsequent subtraction of the recurring nines. These operations seem reasonable, but they also seem undefined. It does not seem valid to do arithmetic on infinitely long decimal fractions. There must be some subtle unstated assumptions about such operations. Reasonable assumptions about transfinite and infinitesimal numbers are often incorrect.

The proof seems valid, especially since there is a valid proof showing that the limit of recurring .99999 is one (this proof does not rely on decimal notation). Without the other proof, I would be nervous about accepting the one involving arithmetic on infinitely long decimal numbers. I would worry about the possibility of such methods leading to erroneous results in another context.

BTW: In base 16, you can similarly prove that recurring .FFFFF equals one. Does this prove that recurring .99999 in decimal equals recurring .FFFFF in hex? For all finite examples, the two recurring fractions are unequal. Note that you can certainly prove that they approach the same limit (namely one).