Let $\{ p_n \}$ be a family of monic orthogonal polynomials associated with a inner product $\langle f,g\rangle = \int_a^b w(x)f(x)\overline{g(x)}dx$. Show that for any monic polynomial of defree $n$, we have that $\langle q,q\rangle \geq \langle p_n,p_n\rangle$.
I feel like it should be an application of some well-known inequalities. I've tried to use Cauchy-Schwarz but haven't got anything.
Any hints or comments are welcome! Thank you.