Proposition 2

Let $X$ be a compact space, and let $H$ be a set of continuous real-valued functions on $X$ such that, given any two functions $u \in H, v \in H$, the functions $\sup(u, v)$ and $\inf(u, v)$ are in $H$. Then a continuous real-valued function $f$ on $X$ can be uniformly approximated by functions belonging to $H$ if and only if, for each real number $\varepsilon > 0$ and each pair $x, y$ of points of $X$, there is a function $u_{x, y} \in H$ such that $|f(x) - u_{x, y}(x)| < \varepsilon$ and $|f(y) - u_{x, y}(y)| < \varepsilon$.

The condition is clearly necessary; let us show that it is sufficient. For each $\varepsilon > 0$, we shall show that there is a function $g \in H$ such that $|f(z) - g(z)| < \varepsilon$ for all $z \in X$. Let $x$ be any point of $X$, and let $H_x$ be the set of all functions $u \in H$ such that $u(x) < f(x) + \varepsilon$. By hypothesis, for each $y \in X$, the function $u_{x, y}$ belongs to $H_x$ and we have $u_{x, y}(y) > f(y) - \varepsilon$. Hence, by Proposition 1, there is a finite number of functions of $H_x$ whose upper envelope $v_x$ is such that $v_x(z) > f(z) - \varepsilon$ for all $z \in X$; on the other hand, we have $v_x(x) < f(x) + \varepsilon$ by the definition of $H_x$; finally, $v_x \in H$ by hypothesis. Proposition 1 therefore shows that there exists a finite number of functions $v_{x_i}$ whose lower envelope $g$ is such that $g(z) < f(z) + \varepsilon$ for all $z \in X$; but since we have $v_{x_i}(z) > f(z) - \varepsilon$ for all $z \in X$ and for every index $i$, we have also $g(z) > f(z) - \varepsilon$ for all $z \in X$. Since $g \in H$ by hypothesis, the proof is complete.