How To Multiply Repeatig Decimals

I was wondering, is it possible to multiply 0.333...*0.666... and get the correct result of 0.222... without first turning the repeating decimals into fractions.

I have played with a couple of ideas

http://i7.photobucket.com/albums/y28.../Equation1.jpg

Is it possible to multiply the 2 summations together?

I have also looked at 0.333...*0.666... = 3(6/10+6/10^2+6/10^3+....)

But I am wodering where to go from here.

Any ideas would be appreciated.

My conclusion is that turning them into fractions then multipying is the simplist way. I do want to see other methods, practical or not.

Thank you for your time