This is not a formal proof, it is more of a demonstrations
You can show by division that $${\frac{{\mathtt{1}}}{{\mathtt{3}}}}$$ = 0.3 repeater.
If you multiply both sides by 3 you get $${\frac{{\mathtt{3}}}{{\mathtt{3}}}}$$ = 0.9 repeater
but $${\frac{{\mathtt{3}}}{{\mathtt{3}}}}$$ = 1 so
1 = 0.9 repeater