Suppose that $(E,d)$ is a locally compact metric space with the metric topology. I am given a particular countable dense subset $S$ of $(C_c(E), \|\cdot\|_{\infty})$ i.e. in the supremum norm. By definition, this means that for all $f \in C_c(E)$, there exists a sequence $f_n \in S$ such that $\|f_n - f\|_{\infty} \to 0$ as $n \to \infty$.
My question is the following : can I automatically deduce the following stronger statement.
For all $f \in C_c(E), f \geq 0$, there exists an increasing sequence of functions $f_n \in S$ such that $f_n \to f$ in $C_c(E)$? (even pointwise convergence will be a useful result). That is, in addition to $f_n \to f$, can I insist that $f_n \geq f_m$ for $n \geq m$, so that $f_n \uparrow f$ as $n \to \infty$ pointwise?
Note that there is a fundamental difference if we are only requiring some $S$ for which this property holds : the most canonical construction of $S$ via the restriction to compact subsets of $E$ which exhaust $E$, satisfies this property. The brief idea of that construction can be found here, and for the resulting $S$, I believe that the assertion in question is true.
However, as a result of another definition, I am only allowed to assume that some $S$ exists, not that it has a particular form. Can I get the same result with arbitrary $S$?
Note : I have asked a similar question over at MathOverflow. The two questions are subtly different in that there, I also insist on a convergence in $L^2(\mu)$ for some measure $\mu$. I assume this question's solution will help me answer it, though.
Note : The question over at MO has an answer. Given $S$ countable, it can be enlarged to another countable set $S'$ , the family of maxima of finite subsets of $S$, and for this family the claim works out, in a "step-function" style proof. Head over there to find out, but I don't want to enlarge $S$ at all for this question.
I probably miss something, but I think here is a counterexample.
Let $E = \mathbb R$ with standard metric. Let $S'$ be any countable dense subset of $C_c(\mathbb R)$. Let $\omega$ we bump function, $\omega(0) = 1$, zero outside $[-1, 1]$ and $|\omega(x)| \leq 1$, and let $\omega_n(x) = \omega(x - n) / n$.
For $f \in S'$, let $f^s$ be number s.t. support of $f$ is subset of $[-f^s + 1, f^s - 1]$.
Then take $S = \{f + \omega_k | f \in S', k \in \mathbb N, k > f^s\}$. As $\omega_k \rightrightarrows 0$, $S$ is also dense in $C_c(\mathbb R)$.
Assume $f_n \rightrightarrows 0$. Then $f_1 = f + \omega_k$ for some $f \in S$, $k \in \mathbb N$. Then $f_1(k) = \frac{1}{k}$. But some $f_m$ should have $|f_m(k)| < \frac{1}{k}$ and thus $f_m(k) < f_1(k)$.