 Solution

For $d=p-q$, all $d(x_i)$ and $d^{\prime }(x_i)$ vanish. Between each consecutive pair of nodes, Rolle's theorem gives one additional zero of $d^{\prime }$. Along with the $n+1$ nodal zeros, this supplies $2n+1$ distinct zeros on the closed interval. Since $d^{\prime }$ is a polynomial of degree at most $2n$, it must be the zero polynomial. Thus $d$ is constant, and $d(x_0)=0$ makes it zero. Hence the interpolating polynomial is unique.

The direct zero count includes the two endpoints; its guaranteed interior count before the degree argument is $2n-1$. After $d^{\prime }=0$ is proved, $d^{\prime }$ vanishes at every point of the open interval, so the stronger interior-zero assertion in the source follows as well. Another immediate uniqueness proof is that $d$ has a double zero at every node, so is divisible by ${\prod }_i(x-x_i{)}^2$ of degree $2n+2$, although its own degree is at most $2n+1$.