I'm recently came across the following theorem:
Let $f$ be a function in $C^{n+1}[a, b]$ and let $p$ be a polynomial of degree $\leq n$ that interpolates the function at $n+1$ distinct points $x_0, \dots x_n \in[a, b]$. Then to every $x \in [a,b]$ there exists a point $\xi_x \in [a, b]$ such that:
$$f(x) - f(p) = \frac{1}{(n+1)!}f^{(n+1)}(\xi_x)\prod_{i=0}^{n}(x - x_i)$$
This seems fine, but later my professor followed with the corollary:
$$||f - p||_{\infty, [a,b]} \leq \frac{|\max \{x_0, \dots x_n\} - \min \{x_0, \dots x_n\}|^{n+1}}{(n+1)!} ||f^{(n+1)}||_{\infty, [a,b]}$$
and I don't understand why we can assume $|\prod_{i=0}^{n}(x - x_i)| \leq |\max \{x_0, \dots x_n\} - \min \{x_0, \dots x_n\}|^{n+1}$. Couldn't $|x-x_i| > |\max \{x_0, \dots x_n\} - \min \{x_0, \dots x_n\}|$ for every $i \in \{0, \cdots n\}$?
For convenience let the points $\{x_i\}$ be ordered, so that $x_0=\min \{x_0, \dots x_n\}$ and $x_n=\max \{x_0, \dots x_n\}$. Then, if $x_0-a>x_n-x_0$ or $b-x_n>x_n-x_0$, there will be points $x\in[a,b]$ such that $|x-x_i|>x_n-x_0$ for all $i$. In that case, the interpolation points are far away from the extremes of the interval $[a,b]$.