True / False: (I) Continuous functions on $[1, 2]$ can be approximated uniformly by a sequence of odd polynomials (i.e., polynomials $p(x) \in \mathbf{R}[x]$ such that $p(-x) = -p(x)$).
(II) Continuous functions on $[1, 2]$ can be approximated uniformly by a sequence of even polynomials (i.e., polynomials $p(x) \in \mathbf{R}[x]$ such that $p(-x) = p(x)$).
Attempt: (II) forms an algebra and satisfy all condition of Stone Weierstrass theorem. So it is true. But what about (I), it not forms an algebra.
Let $f \in C([1,2])$ be arbitrary. Then $\frac{f(x)}{x} \in C([1,2])$ as well so by (II), there exists a sequence $(p_n)_n$ of even polynomials such that $p_n(x) \to \frac{f(x)}{x}$ uniformly. Then $$\|xp_n(x)-f(x)\|_\infty \le 2\left\|p_n(x)-\frac{f(x)}{x}\right\|_\infty \xrightarrow{n\to\infty} 0$$ so $xp_n(x) \to f(x)$ uniformly and $(xp_n(x))_n$ are odd polynomials.