This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds
$$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty \max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)|$$
My lecturer remarked that if the sequence $\left(\| f^{(n)} \|_\infty \right)_{n\in \mathbb N} $ is bounded then the interpolation error goes to zero if we increase the number of sampling points, that implies that
$$\frac{1}{(n+1)!}\max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)| \xrightarrow{n\to\infty} 0$$
I don't quite see why should it be the case. For any given $x_0, \ldots, x_n$, finding the maximum is quite tedious and I'm aware of any way to bound it.
You don't have to find the maximum; a crude upper bound suffices in this case. Since $|x-x_i|\le |b-a|$ for every $i=0,1,\dots,n$, it follows that $$|(x-x_0)\cdots(x-x_n)| \le (b-a)^{n+1}$$ When divided by $(n+1)!$, this goes to zero: the factorial grows super-exponentially.
The above estimate also hints that we can allow certain growth of derivatives, and that the convergence may have something to do with the convergence of the Taylor series of $f$.