TITLE: About a specific argument purporting to show $0.999\dots = 1.0$.
QUESTION [8 upvotes]: I have read the proofs about why $0.9999.... = 1$, which are satisfying. But I can't get the following argument out of my head.
Defining $0.9999....$ : Lets construct a non-terminating but recurring real number n such that all digits before decimal point are zero and all digits after decimal point are 9. 
Comparing $1.0000$ with $0.99999...$
Digit at ones place in $1.0$ (i.e. 1) $\ne$ Digit at ones place in $0.99999$ (i.e. 0)
Digit at tenths place in $1.0$ (i.e. 0) $\ne$ Digit at tenths place in $0.99999$ (i.e. 9). And so on....
Hence, $1.0 =0.9999...$ does not fit with our original definition of $0.9999...$
Can you find the mistake in the argument (other than saying that in-fact $1.0 = 0.9999...$)?
Am I using a incorrect way to define (or perhaps compare) a number (with another)?
Please help me. I am new to analysis. Thanks.

REPLY [3 votes]: The numerals are indeed different. However, that does not mean the numbers they represent are different.
The idea that different things can represent the same number is a familiar one: just think of $1/2$ versus $3/6$ or $1+2$ versus $3$.
I assert the main reason that things like $0.\overline{9} = 1.\overline{0}$ give people trouble is simply because every number that can be represented by a terminating decimal has a unique representation as a terminating decimal, but this property fails to hold when passing from the special case finitely long numerals to the general case infinitely long numerals, coupled with the fact that most real numbers have a unique representation in this fashion.