I am really struggling on this exercise, and don't even know where to start.
(a) Use the fact that $|a|=\sqrt{a^2}$ to prove that, given $\epsilon>0$, there exists a polynomial $q(x)$ satisfying $||x|-q(x)|<\epsilon$ for all $x$ in $[-1,1]$.
(b) Generalize this conclusion to an arbitrary interval $[a,b]$.
I recognize that this is the conclusion of the Weierstrass approximation theorem. But I don't know how to use it.
Thank you so much in advance for your help!