I've spent more time that I'd like to admit working out this bound.
Online, I've seen that the remainder $R(x) = f(x) - L(x)$ in Lagrange interpolation is bounded by
$$| R(x) | \leq \frac{(x_n - x_0)^{n + 1}}{(n + 1)!} \max_{x_0 \leq \xi \leq x_n} f^{(n + 1)} (\xi).$$
I've tried to prove it, but gotten nowhere. Here, $f$ is an $n + 1$ times continuously differentiable function on a compact interval $I$, and $L(x)$ is the interpolating polynomial of $f$ with respect to the $n + 1$ nodes $\lbrace x_0, \dots, x_n \rbrace \subset I$.
My approach was to note that by applying Rolle's theorem many times one can deduce that
$$R(x) = \frac{f^{(n + 1)} (\xi)}{(n + 1)!} \ell (x).$$
Where $\ell (x) = \prod_{0 \leq k \leq n} (x - x_k)$ and $x_0 \leq \xi \leq x_n$ is a point that depends on $x$. Now, I believe that the inequality above is proven if I can show that
$$| \ell (x) | \leq (x_n - x_0)^{n + 1}.$$
I tried to find the critical points of this polynomial, but this doesn't seem to work out.