(a) State Chebyshev's equal ripple criterion.

(b) Let $f:[-1,1] \rightarrow \mathbb{R}$ be defined by

$$f(x)=\cos 4 \pi x$$

and let $g$ be a polynomial of degree 7 . Prove that there exists an $x \in[-1,1]$ such that $|f(x)-g(x)| \geqslant 1$.