On a paracompact space, there is a continuous positive function with given local bounds Let $X$ be a paracompact hausdorff space. Let $\mathcal{C}$ be a locally finite family of subsets of $X$. For $C\in \mathcal{C} $, let $\delta(C)$ be a positive real number.
I want to show that there is a continuous function $f\colon X \to (0, \infty)$ such that $f(C)\subseteq (0,\delta(C)]$.
I suspect that one can prove this by using partitions of unity, but have had no idea as to which open covering to use. 
 A: For each $x$, choose an open neighbourhood $U_x$ of $x$ that intersects only finitely many members of $\mathcal{C}$. Then let
$$\eta(x) = \min \{ \delta(C) : C \in \mathcal{C}, U_x \cap C \neq \varnothing\}.$$
Let $\{ \varphi_x : x \in X\}$ a partition of unity subordinate to the cover $\{U_x : x \in X\}$(1) and set
$$f = \sum_{x \in X} \eta(x)\cdot \varphi_x.$$

(1) That is, $\varphi_x \colon X \to [0,1]$ is continuous, $\operatorname{supp} \varphi_x \subset U_x$, and the family $\{ \operatorname{supp} \varphi_x : x \in X\}$ is locally finite. Usually, a lot of the $\varphi_x$ will be identically $0$.
In case it's not yet known that on a paracompact Hausdorff space for every open cover $\mathscr{U}$, whether locally finte or not, there is a partition of unity subordinate subordinate to $\mathscr{U}$: Let $\mathscr{W}$ a locally finite open refinement of $\mathscr{U}$. Let $\{ \psi_W : W \in \mathscr{W}\}$ a subordinate partition of unity (I assume the result is known for locally finite open covers). Let $\tau \colon \mathscr{W} \to \mathscr{U}$ a function with $W \subset \tau(W)$ for all $W$, and set $$\varphi_U = \sum_{\tau(W) = U} \psi_W.$$
