= Solution
The real <Stone-Weierstrass theorem> states that if $X$ is a compact Hausdorff space and $A\subseteq C(X,\mathbb R)$ is a subalgebra containing the constants and separating points of $X$, then $A$ is dense in $C(X,\mathbb R)$ in the <uniform norm>.
The <Tietze extension theorem> states that if $X$ is a normal topological space, $F\subseteq X$ is closed, and $f:F\to\mathbb R$ is continuous, then $f$ extends to a continuous function on $X$. In its bounded form, an $f$ taking values in $[-M,M]$ has an extension with the same range bound. The underlying separation step is the <Urysohn lemma>.
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