 Solution

Given $ϵ>0$, the Weierstrass approximation theorem supplies a polynomial $Q$ with $\Vert f-Q{\Vert }_{\infty }<ϵ$. For all sufficiently large $n$, the quadrature is exact on $Q$. Positivity of the weights and ${\sum }_j{A}_j=\int r$ then giveLetting $ϵ\downarrow 0$ proves convergence.