Also known as 0.999999........
It is a little known, yet very important fact that the Arab numeral system, which is universally used nowadays, is not unambiguous. Any finite decimal fraction can be represented by an infinite decimal fraction by decrementing the last digit and adding a trail of 9's. Thus, the number 0.9999... is the same number as the number 1. This also holds for numbers like 0.4, which can be represented by 0.39999999...
Note that this is not some kind of limiting property! It is not so that the number 0.99999... 'approaches' 1. It is, most definitely the same number. In some mathematical proofs, like the proof that the set of real numbers is uncountable, special measures have to be taken to eliminate this ambiguity. That is, the existence of both 0.99999.. and 1 seems to break the proof, but since both are identical, it does not.
Casual readers may stop here. Pedants read on..
It has been argued by some that this is in fact a limiting property. To see why this is utterly nonsensical, let us review what the definition of a limit is:
Let f be a function from R to R. If, for some a there exists an L such that for every ε > 0 there exists a δ such that:
|x - a| < δ ⇒ |f(x - a) - L| < ε
..then L is defined as the limit of f in a.
From this definition it should be unambiguously clear that the concept of a limit applies to functions and not to constants. You could argue that a number is a constant function, but then you would have your definitions seriously mixed up. To illustrate that last point: even if we do not define the concept of 'function' at all, 0.99999... is still 1. And, since the definition of a limit requires the definition of the function concept, this implies that the limiting construct need not be defined either for 0.99999.. to be 1 QED.
I'll Fitch it for you if you want! ;-)
Note: This fact is not a theorem! It is just a commonly accepted sloppy notation of "..." meaning ad infinitum and, worse, that a series like this is actually equivalent to a number. Because of this there is no 'proof' and so this and the 'proofs' in that are exercises in futility.
..if you refuse to understand this, I cannot help you. Stop msging me. You are actually wrong and not because I am being pedantic. Please read a book before you downvote me.
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