How prove this nice limit $\lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$ let $f(x)$ is Continuous on $[0,1]$, and such $f(0)=f(1)$,and if $a$ is irrational number.
show that

$$\lim_{n\to+\infty}\dfrac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$$

where $\{ka\}=x-[x]$,and $[x]$ is 
is the largest integer not greater than $x$
This problem  is from this ( problem 8)http://wenku.baidu.com/view/a643e6c26137ee06eff91855.html
and I find this Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
Have without Trigonometric series  methods? because  this problem is Freshman exam questions

Let see “Concrete mathematics : a foundation for computer science” book, chapter 3, section 3.5, pages 87–89.
