Positive weights of Gaussian quadrature ($A_j=\int {\ell }_j^2>0$)

For an $n$-node Gaussian quadrature rule exact through degree $2n-1$, the cardinal Lagrange interpolation polynomial ${\ell }_j$ has square of degree $2n-2$. Exactness gives $A_j=\int {\ell }_j^2>0$ for a positive integration weight. This proof turns polynomial exactness directly into positivity.