Interpolation property of weighted least-squares polynomial approximation ($\#\{x:f(x)=p_n(x)\}\ge n+1$)

Let $f$ be a real continuous function on a nondegenerate compact interval and let $p_n$ be its best approximation among polynomials of polynomial degree at most $n$ for a positive integrable weighted inner product. Then $f-p_n$ has at least $n+1$ distinct zeros. Indeed, if it had at most $n$, multiply it by the polynomial with simple zeros at precisely its interior sign changes. The product has constant nonzero sign away from finitely many zeros, so its weighted integral is nonzero, contradicting orthogonality of $f-p_n$ to all polynomials of polynomial degree at most $n$.