= Integer polynomial approximation away from zero and one
On $[a,b]\subset(0,1)$, the <uniform closure> of integer-coefficient <polynomials> contains $1/2$, since $\sum_{j=0}^Nx(1-2x)^j\to1/2$ uniformly. This closure is a ring, so it contains all dyadic constants and then all real constants. It contains $x$ and therefore every real-coefficient polynomial. The <Stone-Weierstrass theorem> makes it all of $C([a,b])$. On an interval including zero this fails, since the value of an integer-coefficient polynomial at zero is an integer.
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