One way to argue that 0.99999… is equal to 1 is the following:
1/3=0.33333…
Multiply the above equation by 3,
1=0.99999…
Is the above convincing?
If you are still not convinced, we can let x=0.99999…
Then, 10x=9.99999…
10x-x=9.99999…-0.99999…=9
9x=9
Hence, x=1.
Grade 4 Decimals & Fractions (Kumon Math Workbooks)
There are also more advanced methods of proving 0.999…=1, listed here on Wikipedia. (http://en.wikipedia.org/wiki/0.999…) Some of the techniques include Infinite series and sequences, Dedekind cuts, and Cauchy sequences.
