Suppose that $x_1,x_2,\ldots x_n$, are the roots of a degree $n$ orthogonal polynomial $Q_n(x)$ on $[a,b]$. Consider the corresponding Gaussian Quadrature
$$\int_a^b f(x) dx \approx \sum_{j=1}^nA_jf(x_j)$$
where $A_j=\int_a^b l_j(x) dx$ for Lagrange basis polynomials $l_j$ associated with $x_1,x_2,\ldots x_n$.
Show that $|\int_a^b f(x) dx-\sum_{j=1}^nA_jf(x_j)| \leq 2(b-a)\min_{p \in P_{2n-1}}||f-p||_{C[a,b]}$
I know how to derive the error estimates when $f \in C^{2n}[a,b]$ but without that assumption, I don't have any estimates on $f$ to begin with. I suppose we have to use the fact that Gaussian Quadrature is exact for all $p \in P_{2n-1}$ but I don't know where to begin.
How about taking any polynomial $p\in P_{2n-1}$, then
$$\int_a^b f(x) dx - \sum_{j=1}^n A_j f(x_j) = \int_a^b ( f(x) - p(x) )dx + \int_a^b p(x) dx - \sum_{j=1}^n A_j f(x_j) $$
Since the quadrature rule is exact for $p$, it becomes
$$\int_a^b ( f(x) - p(x) )dx + \sum_{j=1}^n A_j (p(x_j) - f(x_j) )\le |a-b||f-p|_{\infty} + \sum_{j} A_j |p-f|_{\infty} = 2|a-b||f-p|_{\infty}.$$
Of course $A_j > 0$ is needed, this is a conclusion from Gaussian quadrature.