Cylinder-function density in Cantor space (${\overline{\mathrm{span}\{1_C:C\text{ a cylinder}\}}}^{\Vert \cdot {\Vert }_{\infty }}=C(\Omega )$)

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.