I am currently trying to prove bounds for some polynomial interpolants (Lagrange to be precise) and am having problems proving the following:
$$ \left|\left| f(x) - p_n(x) \right|\right|_{\infty} \leq C_nh^{n+1}\left| \left|f^{(n+1)}(\xi_x)\right| \right|_{\infty} $$
where $f(x)$ is the function to be appoximated, $p_n(x)$ is the interpolant of degree $n$, $h$ is the uniformly spaced time step $\left( =\frac{b-a}{n} \right)$ and $\xi_x\in [a,b]$ (The function is to be approximated between the values $a$ and $b$).
I have taken the following for granted:
$$ f(x)-p_n(x) = \frac{w_{n}(x)}{(n+1)!}f^{(n+1)}(\xi_x) $$
where
$$ w_n(x) = \prod^{n}_{k=0}(x-x_k) $$
Now in an effort to compute the bounds:
\begin{align} \left|\left| f(x)-p_n(x) \right|\right|_{\infty} &= \max_{x\in[a,b]} \left| f(x) - p_n(x)\right|\\ &= \max_{x\in[a,b]}\left|\frac{w_{n}(x)}{(n+1)!}f^{(n+1)}(\xi_x)\right|\\ &\leq \max_{x\in[a,b]}\left|\frac{w_{n}(x)}{(n+1)!}\right|\max_{\xi_x\in[a,b]}f^{(n+1)}(\xi_x) \end{align}
However, looking through An Introduction to Numerical Methods and Analysis (Epperson, 2002), the following steps have been made
$$ \max_{x\in[a,b]}\left|\frac{w_{n}(x)}{(n+1)!}\right| = C_nh^{n+1} $$
due to uniform spacing, and
$$ C_1=\frac{1}{8}, \ \ C_2=\frac{1}{9\sqrt{3}}, \ \ C_3=\frac{1}{24}, $$
I do not understand this final step and in particular where these values came from. I would appreciate any hints to point me towards the right direction.