I can't see the following:
Let $\phi_0,\phi_1,...$ be a sequence of orthonormal polynomials on an interval $[a,b]$ w.r.t positive weight function $w(x)$. Let $x_1,..., x_n$ be the $n$ zeros of $\phi_n(x)$. Prove that $$ L_j(x)L_k(x)=\phi_n(x)q(x)\quad\textrm{for }j \neq k \quad\textrm{where} \quad L_j(x)=\prod_{\substack{k=1 \\ k \neq j}}\frac{(x-x_k)}{(x_j-x_k)}\quad 1\leq j \leq n. $$ and $q$ is a polynomial of degree at most $k-1$.
I need that fact to prove the orthoganality of Lagrange polynomials.
Thanks.