Sam: Variations on this argument have occurred several times now.

If you tell me how many nines we are talking about, I will tell you what 1 - .99999... is equal to, and I will estimate the reciprocal. I will almost always insist on being picky, because I believe in avoiding sloppy terminology and sloppy methods. In practice, I will use one as a replacement for .999999 and 1/3 as equivalent to .333333, but I will not accept statements like " recurring .33333 equals 1/3," except as convenient shorthand notation.

This thread started with a proof that recurring .999999 = 1.0000 using some questionable operations on "infinitely long" decimal numbers. I objected to the proof for various reasons and suggested viewing the recurring .99999 as a geometric series. It is fairly easy to prove that the limit of the series is one, without using questionable arithmetic operations on infinitely long decimal numbers.

I agree with Noble
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".
Limits are defined by terminology similar to the following.
The number a will be called the limit of the sequence X1, X2, X3 . . . Xn . . . provided that, given any positive number h, no matter how small, there exists a corresponding term of the sequence Xn such that every succeeding term of the series lies between the numbers a-h and a+h.
The wording is slightly different for the limit of a function and the limit of the sum of a series, but the idea is always the same. As some number or variable grows without bound, some value gets closer to the limiting value.

The formal definitions use terms like increases indefinitely or there exists or grows without bound. The formal definitions refer to an extremely small difference between the value and its limit. They avoid referring to infinity and avoid the statement that the value equals its limit.

The use of the limit as a value is viewed as a convenience which avoids lengthy terminology.

Every serious mathematical text I have ever read avoids claiming that a function equals its limit. When not dealing with transfinite numbers ala Cantor, they also avoid using the term infinity. I have a book (Fundamentals of Mathematics by Moses Richardson) which specifically warns against the use of such a term.
The notation "Limit(Xn) = a as n —>lazyeight" is commonly used, but it often misleads the unwary student into believing that there is a peculiar number called infinity which n approaches. This is, of course, not true.
In a later chapter of this book
. . .It would be even more misleading to write 1/0 = infinity, although some books do. . . .they do not mean that infinity is a number which you obtain by dividing 1 by zero. As we have seen 1/0 is a meaningless symbol and infinity is not a number. "Infinity" in this sense is merely a way of describing the manner in which certain function behave.
In a footnote it is mentioned that historically the statement 1/0 = infinity was taken literally before mathematicians were familiar with modern mathematical logic.