Write $C[a,\infty)$ for the set of all real-valued continuous functions defined on $[a,\infty)$.
Let $\emptyset \neq K \subset C[a,\infty)$ which satisfies $\forall f, g \in K$, we have $h \in K$ where $h(x) = min\{f(x),g(x)\} \forall x \in [a,\infty)$.
Let $g \in C[a,\infty)$. Suppose that $g(x) =$ inf$\{f(x): f \in K; f \geq g\} \forall x ∈ [a,\infty)$.
Can we approximate such $g$ by functions in $C[a,\infty)$?
That is $\forall \epsilon > 0$, does there exists some $f \in K$ such that $|f(x) - g(x)| < \epsilon \forall x \in [a,\infty)$?
My idea is no because the domain is non-compact.
If $g$ is unbounded, it looks false. Let for example $K$ be the set of piecewise linear continuous functions with a finite number of pieces and let $g(x) = \sqrt{x}$. Clearly $g$ cannot be approximated uniformly by elements of $K$.
In the bounded case, it looks false as well, take for $K$ the set of piecewise linear continuous bounded functions with a finite number of pieces and let $g(x)=\sin(x)$. The same problem occurs.