First, to show that rational => repeating decimal expansion, remember that long division is how you get from a rational number to its decimal expansion. If r=a/b, then there are only b possible remainders at each step. What happens when the same remainder comes up a second time (after you've gone through all the digits in a)? I'll let you work on the other direction.

Ok well I guess my intuition was bad. I made a few search on books.google.com and found a couple of proofs. One is pretty easy and I actually though of that solutions before but didn't know how to generalise it. My only problem now is that I'm not familiar with one notation in the proofs.

I don't understand the reprensetation of the period, so in this image the first line where it says x= a_n ...