We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$
$P(x)$ is the interpolation polynomial of $f(x)$ where $P(x_i)=f(x_i)$ for all $i$.
We define $h=\frac{1}{n}$.
Show that for any $x\in [0,1]$ we get $|(x-x_0)(x-x_1)\dots(x-x_n)|\leq \frac{n!h^{n+1}}{4}$
I'm not entirely sure where to begin. Perhaps induction is a good way to start? I'd appreciate any help I can get.
If your points are $0, h, 2h, ..., (n - 1) h, 1$, this means that the largest value of your expression happens when $x =0$. Reformulate as a statement when the points are $0, 1, 2, \ldots, n$, that will be easier to handle.