fractional part and integration

Let $\displaystyle \{x\}$ be the positive fractional part of real number $\displaystyle x$, for example, $\displaystyle \{\pi\}=0.1415926...$ .

Prove that the set $\displaystyle S=\{\{nv\}:n\in Z^+\}$ is dense in [0,1] if $\displaystyle v$ is positive irrational number.

And for any $\displaystyle f(x)\in C[0,1]$(the class of all real-valued continious function on [0,1]), and any positive irrational number $\displaystyle v$, we have

$\displaystyle \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(\{nv\} )=\int_0^1f(x)dx$.

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!