If you have a problem with that proof (Although I don't see one):
1/3 = 0.3*recurring
3/3 = 1
3/3 = (1/3) * 3 = (0.3*recurring) * 3 = 0.9*recurring
Therefore:
1 = 0.9*recurring
1/3 = 0.3*recurring
3/3 = 1
3/3 = (1/3) * 3 = (0.3*recurring) * 3 = 0.9*recurring
Therefore:
1 = 0.9*recurring