Given $n+1$ real points $x_{0}<x_{1}<\cdots<x_{n}$, define the Lagrange cardinal polynomials $\ell_{i}(x), i=0,1, \ldots, n$. Let $p(x)$ be the polynomial of degree $n$ that interpolates the function $f \in C^{n}\left[x_{0}, x_{n}\right]$ at these points. Express $p(x)$ in terms of the values $f_{i}=f\left(x_{i}\right)$, $i=0,1, \ldots, n$ and the Lagrange cardinal polynomials.

Define the divided difference $f\left[x_{0}, x_{1}, \ldots, x_{n}\right]$ and give an expression for it in terms of $f_{0}, f_{1}, \ldots, f_{n}$ and $x_{0}, x_{1}, \ldots, x_{n}$. Prove that

$$f\left[x_{0}, x_{1}, \ldots, x_{n}\right]=\frac{1}{n !} f^{(n)}(\xi)$$

for some number $\xi \in\left[x_{0}, x_{n}\right]$.