Let $[a, b]$ be the smallest interval that contains the $n+1$ distinct real numbers $x_{0}, x_{1}, \ldots, x_{n}$, and let $f$ be a continuous function on that interval.

Define the divided difference $f\left[x_{0}, x_{1}, \ldots, x_{m}\right]$ of degree $m \leqslant n$.

Prove that the polynomial of degree $n$ that interpolates the function $f$ at the points $x_{0}, x_{1}, \ldots, x_{n}$ is equal to the Newton polynomial

$$p_{n}(x)=f\left[x_{0}\right]+f\left[x_{0}, x_{1}\right]\left(x-x_{0}\right)+\cdots+f\left[x_{0}, x_{1}, \ldots, x_{n}\right] \prod_{i=0}^{n-1}\left(x-x_{i}\right)$$

Prove the recursive formula

$$f\left[x_{0}, x_{1}, \ldots, x_{m}\right]=\frac{f\left[x_{1}, x_{2}, \ldots, x_{m}\right]-f\left[x_{0}, x_{1}, \ldots, x_{m-1}\right]}{x_{m}-x_{0}}$$

for $1 \leqslant m \leqslant n .$