Lets say that we have the well known Gauss-Chebyshev Quadrature rule of the first kind i.e,
$\int^1_{-1}f(x)\frac{1}{\sqrt{1-x^2}}dx= \sum^n_{k=1} w_kf(x_k) +R_n(f) $, where the nodes $x_k$ are $x_k=\cos(\frac{2k-1}{2n})$, and the weights are all equal $w_k=\frac{\pi}{n}, \quad k=1,2,...,n$
and we want to show that the quadrature sum $Q_n(f)=\sum^n_{k=1} w_kf(x_k)$ is a Riemannian sum. We already know that the Quadrature rule converge for Riemann integrable functions but I dont want to use that. How can I show that the sum $Q_n(f)=\sum^n_{k=1} w_kf(x_k)$ is a Riemannian sum?