Let be the positive fractional part of real number , for example, .

Prove that the set is dense in [0,1] if is positive irrational number.

And for any (the class of all real-valued continious function on [0,1]), and any positive irrational number , we have

.

I know how to do this problem, but i can't write down the formal solution and some details completely.

I need a complete and elaborate solution, thanks in advance!