Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 1 / b / Solution

For $n\ge 1$, set $p=2^{1-n}{T}_n$. It is monic by part (a), and its supremum norm on $[-1,1]$ is $E=2^{1-n}$. At the $n+1$ successive extreme points $x_j=\cos (j\pi /n)$ it alternates between $E$ and $-E$.

Suppose a monic degree-$n$ polynomial $q$ had $\Vert q{\Vert }_{\infty }<E$. At each $x_j$, the difference $p-q$ would have the sign of $p$. These signs alternate, so the intermediate value theorem forces a root in each of the $n$ successive gaps. But the leading monic terms cancel, making $\mathrm{deg}(p-q)\le n-1$, a contradiction. ThereforeThis is the monic monic Chebyshev extremal polynomial. The positive-degree restriction matters: at $n=0$ the printed scaling gives the constant two, while the only monic constant is one, so that version is false. The customary theorem uses $n\ge 1$.