Although some people find it counterintuitive, the decimal expansions and represent the same real number.
These "proofs" can help give insight, but be careful; a similar technique can "prove" that . They work in this case because the series corresponding to is absolutely_convergent.
The real numbers are dense, which means that if , there must be some number in between. But there's no decimal expansion that could represent a number in between and .
This is a more formal version of the first informal proof, using the definition of decimal notation.
is the decimal expansion where every digit after the decimal point is a . By definition, it is the value of the series . This value is in turn defined as the limit of the sequence . Let denote the th term of this sequence. I claim the limit is . To prove this, we have to show that for any , there is some such that for every , .
Let's prove by induction that . Since is the sum of {$0$ terms, , so . If , then
So for all . What remains to be shown is that eventually gets (and stays) arbitrarily small; this is true by the Archimedean_property and because is monotonically decreasing.
These arguments are used to try to refute the claim that . They're flawed, since they claim to prove a false conclusion.
Decimal expansions and real numbers are different objects. Decimal expansions are a nice way to represent real numbers, but there's no reason different decimal expansions have to represent different real numbers.
Decimal expansions go on infinitely, but no farther. doesn't represent a real number because the is supposed to be after infinitely many s, but each digit has to be a finite distance from the decimal point. If you have to pick a real number to for to represent, it would be .
The sequence gets arbitrarily close to , so its limit is . It doesn't matter that all of the terms are less than .
There are infinitely many s in , so when you shift it over a digit there are still the same amount. And the "decimal expansion" doesn't make sense, because it has infinitely many digits and then a .