Let $f:[a,b]\rightarrow \mathbb{R}$ be a riemann integrable function. For $n\in \mathbb{N}$ let $a=x_0<x_1<...<x_n=b$ be a partition of the interval with $x_k=a+\frac{k}{n}(b-a)$ and $\xi_k\in [x_{k-1},x_{k}]$. Furthermore let a sequence of functions be given by: $$f_n(x)= \begin{cases} f(\xi_k),\ x\in (x_{k-1},x_{k}); k\in \{1,...,n \} \\ f(x), \ x=x_k; \ k\in \{0,...,n \} \end{cases} $$ Prove that $\lim_{n\rightarrow \infty} \int_a^bf_n(x)dx=\int_a^bf(x)dx$ and also that $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}f(\xi_k)=\int_a^bf(x)dx. $$
This is probably the most dense question I've come across in analysis and I'm not sure where to begin exactly so I'd appreciate some tips. In a different question I managed to prove that if arbitrary integrable sequence $f_n$ converges uniformly to a $f$ then the equality involving the integrals holds true. So I was thinking of looking at $f_n$ to see if it converges uniformly and then maybe go on to see if the second equality holds but I'm not sure.