 Solution

Let $\omega (t)={\prod }_{r=0}^n(t-x_r)$. For $x$ not one of the nodes, put $c=(u(x)-p_n(x))/\omega (x)$ and defineThis function has zeros at the $n+1$ interpolation nodes and at the additional point $x$. Repeated Rolle's theorem gives a point $\eta$ between the outermost of these points such that $F^{(n+1)}(\eta )=0$. Since $p_n^{(n+1)}=0$ and ${\omega }^{(n+1)}=(n+1)!$,Substitution yieldsAt an interpolation node both sides are zero, so any $\eta \in [-1,1]$ gives the identity there. The assumed $C^{n+1}$ regularity permits all applications of Rolle's theorem.