Approximation of continuous functions in integral

Question

$\textbf{Problem}$ Let $\{x_n\}$ be a squence in $[0,1]$ such that for any $h\in \mathbb{Z}$ \begin{align*} \lim _{N\rightarrow \infty}\frac{1}{N}\sum_{n=1}^N e^{2\pi i hx_n}= \begin{cases} 1 \quad h=0\\ 0 \quad h\neq 0 \end{cases} \end{align*} Show that for any continuous functions $f$ on $[0,1]$ with $f(0)=f(1)$ \begin{align*} \lim_{N\rightarrow \infty} \frac{1}{N} \sum_{n=1}^{N} f(x_n)=\int_0^1 f(x) \; dx \end{align*}

I thought this problem is similar with the Weierstrass approximation theorem.. However, I stuck how to use the first condition...

Any help is appreciated...

Thank you!!

Answer 1

Hint: Show it is true if $f$ is a trig. polynomial, and then show the general $f$ is the uniform limit of trig. polynomials.