Lemma 1

For each real number $\varepsilon > 0$ and each compact interval $I \subset \mathbf{R}$ there exists a polynomial $p(t)$ with no constant term such that $|p(t) - |t|| \leq \varepsilon$ for all $t \in I$.

It is enough to prove the lemma for an interval of the form $I = [-a, +a]$ and hence, replacing $t$ by $at$, for the interval $I = [-1, +1]$. Since $|t| = \sqrt{t^2}$, Lemma 1 is a consequence of the following result: