Let $(p_n)_{n\in\Bbb N}$ be the orthonormal sequence of polynomials associated to a tempered weight $w$ on an interval $I$ (so that for example, $\deg(p_n)=n$). Show that $p_n-\lambda p_{n-1}$ has $n$ distinct roots in $I$ for any real $\lambda$.
I'm stuck at how to prove this.
My attempt at a proof: Let $l_n:=p_n-\lambda p_{n-1}$ and $r_1,..r_k$ be the roots of odd multiplicity in I. Let $q=(t-r_1)...(t-r_k)$ . Then $deg(q)=k≤n$. Note that $l_nq$ is either nonpositive or non negative. Hence $<l_n,q>\neq 0$. Since $l_n\in \{p_0,p_1,..,p_{n-2}\}^{perp}$. This implies that $deg(q)≥n-1$. This is where im stuck. Any helps or hints?
After this I have to show that the roots of $p_n$ and $p_{n-1}$ are interlaced. Any hints for that?