Source: A Course in Modern Analysis and Its Applications by Cohen page 196
As in the image, we can see that $(P_a-P_c)(x_n)$ alternates between $\leq 0$ and $\geq 0$.
If we make $(P_a-P_c)(x_n)>0$ whenever $(P_a-P_c)(x_n)\geq 0$ in the image attached and $(P_a-P_c)(x_n)=0$ whenever $(P_a-P_c)(x_n)\leq 0$ in the image attached, we should get a polynomial equal to $0$ at $\geq \frac{r}{2}$(if even) or $\geq\frac{r}{2}+1$ (if odd) points.