Lemma 1

*If Y is a locally compact space with a countable base, the space $\mathscr{H}(Y)$ contains a countable dense subset. More precisely, there exists in $\mathscr{H}(Y)$ a countable subset S consisting of functions $\geq 0$, such that, for every function $f \geq 0$ of $\mathscr{H}(Y)$, there exists a sequence of functions $f_n \in S$ ($n \geq 0$) that converges uniformly to $f$ and is such $f_n \leq f_0$ for all $n \geq 0$.

For, Y is the union of an increasing sequence $(U_n)$ of relatively compact open sets such that $\overline{U}_n \subset U_{n+1}$ for all $n$ (GT, I, §9, No. 9, Prop. 15); the space $\mathscr{H}(Y)$ is the union of the increasing sequence of Banach spaces $\mathscr{H}(Y, \overline{U}_n)$, and each of the latter is known to be separable (GT, Ch. X, §3, No. 3, Th. 1). Let $S'_n$ be a countable dense set in $\mathscr{H}(Y, \overline{U}_n)$, $S_n$ the set of functions $\varphi^+$ for $\varphi \in S'_n$, and $u_n$ a function in $\mathscr{H}(Y, \overline{U}_{n+1})$, with values in $[0, 1]$ and equal to 1 on $U_n$. We take for S the union of the $S_n$ and the set of $m u_n$ for $m$ and $n$ integers $\geq 0$. For every function $f \geq 0$ of $\mathscr{H}(Y)$, $f$ has support contained in one of the $U_n$, hence is the uniform limit of a sequence of functions $f_p \in S_n$ ($p \geq 1$). These functions $f_p$ are uniformly bounded by a positive integer $m$, and it suffices to take $f_0 = m u_n$.*