Which is bigger, 0.9999999…. or 1?

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.