(a) State Chebychev's Equal Ripple Criterion.

(b) Let $n$ be a positive integer, $a_{0}, a_{1}, \ldots, a_{n-1} \in \mathbb{R}$ and

$$p(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} .$$

Use Chebychev's Equal Ripple Criterion to prove that

$$\sup _{x \in[-1,1]}|p(x)| \geqslant 2^{1-n}$$

[You may use without proof that there is a polynomial $T_{n}(x)$ in $x$ of degree $n$, with the coefficient of $x^{n}$ equal to $2^{n-1}$, such that $T_{n}(\cos \theta)=\cos n \theta$ for all $\theta \in \mathbb{R}$.]