Lattice approximation from two-point approximation

Suppose a family $A\subseteq C(K,\mathbb{R})$ on a compact space is closed under pointwise maximum and minimum. If a target $f$ can be approximated arbitrarily closely at each pair of points by a member of $A$, it can be approximated in the supremum norm. For a fixed first point, take finitely many pairwise approximants whose neighborhoods cover $K$ and take their maximum: it lies above $f-\epsilon$ everywhere and below $f+\epsilon$ near that first point. Take finitely many such first-point neighborhoods covering $K$ and take the minimum of their maxima. The result is within $\epsilon$ of $f$ everywhere. Starting with tolerance $\epsilon /2$ ensures a strict uniform approximation error smaller than $\epsilon$.