One simple proof which I'm surprised hasn't been brought up is to consider the difference between the left hand and the right hand sides of the equation.

If you have one decimal, (.9 vs 1) then the difference is 0.1
With two, it is 0.01
With n, it is 10^(-n)

The question is: How many decimals do we have? Well, since it is repeating FOREVER, we take the limit as n approaches infinity. As n approachs infinity, 10^(-n), and hence the difference, approaches zero.

If the difference between the numbers is zero, they are the same.

**POOF**

For my next trick, I need a volunteer from the audience...