= Cylinder-function density in Cantor space
{title2=$\overline{\operatorname{span}\{1_C:C\text{ a cylinder}\}}^{\|\cdot\|_\infty}=C(\Omega)$}
A <continuous function> on <Cantor space> is uniformly continuous. Replacing it by one constant on each length-$n$ prefix <cylinder set> changes it by at most its oscillation on sets of diameter $2^{-n}$. This tends uniformly to zero. Each approximant is a finite linear combination of continuous <indicator functions>, proving density in the <supremum norm>.
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