For each $\epsilon>0$ there is a polynomial $p\colon\mathbb{C}\to\mathbb{C}$ such that $|p(x)-|x||<\epsilon$ for all $x\in[-r,r]$.

Question

I'm looking for an elementary proof (e.g. no Stone-Weierstrass) of the following result:

Fix an $r\geq0$. Then for each $\epsilon>0$ there exists a polynomial $p\colon\mathbb{C}\to\mathbb{C}$ such that $|p(x)-|x||<\epsilon$ for all $x\in[-r,r]$.

I have seen a proof where they analyzed the Taylor series of $[0,1]\ni x\mapsto(1-x)^{1/2}$ in the case $r=1$ (if I remember correctly), but I did not really understand it.

Are there any elementary (elegant) ways to tackle this problem? Any help would be greatly appreciated!