I'm reading a paper (""A simple proof in Monge-Kantorovich duality theory" by D. A. Edwards), in which the following Lemma is proved. I understand the Lemma and its proof, but not the author's conclusion. Maybe someone can help?
Lemma: Let $Z$ be a completely regular space, $h: Z \to [-\infty, \infty)$ upper semicontinuous with $h\leq0$, $z_0 \in Z$ and $t\in \mathbb{R}$ such that $h(z_0)<t$. Then there exists $g \in \mathcal{C}_b(Z)$ such that $h\leq g\leq0$ and $g(z_0)<t$.
The proof basically only uses the completely regular property and the fact that $h\leq 0$.
The author claims: "From this Lemma it follows immediately that there exists a decreasing net $(h_\alpha)$ in $\mathcal{C}_b(Z)$ (real-valued, bounded, continuous functions) that has pointwise limit $h$ and is such that $h_\alpha \leq 0 ~ \forall \alpha$."
By taking $\alpha=t$ I would understand how to construct a net converging at one point $z_0$. But how does it converge pointwise, i.e. for all $z \in Z$, and how is this net decreasing, i.e. $h_{\alpha_1} \geq h_{\alpha_2}$ for $\alpha_2 > \alpha_1$?