= Integer-coefficient polynomial approximation
{title2=$\overline{\mathbb Z[x]}^{\|\cdot\|_\infty}=\{f\in C[0,1]:f(0),f(1)\in\mathbb Z\}$}
The <uniform approximation> closure of the integer-coefficient <polynomials> on $[0,1]$ consists precisely of the <continuous functions> taking <integer> values at both endpoints. Necessity follows because evaluations at zero and one are <integers> and convergent <integer> sequences are eventually constant. For sufficiency, the <rounded Bernstein polynomial> has <integer> monomial <coefficients> and differs uniformly by a vanishing amount from the <Bernstein polynomial>, which converges uniformly to the target.
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