State and prove the Stone-Weierstrass theorem for real-valued functions. You may assume that the function $x \mapsto|x|$ can be uniformly approximated by polynomials on any interval $[-k, k]$.

Suppose that $0<a<b<1$. Let $\mathcal{F}$ be the set of functions which can be uniformly approximated on $[a, b]$ by polynomials with integer coefficients. By making appropriate use of the identity

$$\frac{1}{2}=\frac{x}{1-(1-2 x)}=\sum_{n=0}^{\infty} x(1-2 x)^{n}$$

or otherwise, show that $\mathcal{F}=\mathcal{C}([a, b])$.

Is it true that every continuous function on $[0, b]$ can be uniformly approximated by polynomials with integer coefficients?