Showing that the limit exists within the definition of set function

Question

I would like a hint for the following problem. Thanks for any help in advance.

Let $\mathcal{C}$ be a family of subsets of $\Omega$ (including $\emptyset$) and let $\tau : [0,\infty) \rightarrow [0,\infty] $ be a function with $\tau(0) = 0$. Define the set function $\mu$ on $2^{\Omega}$ by $\mu(A) = \lim_{\delta \downarrow 0} \inf \{\sum_{i=1}^{\infty} \tau(diam(C_{i}) : A \subset \cup_{i=1}^{\infty} C_{i}, C_{i} \in \mathcal{C}, diam(C_{i}) \leq \delta$} for all $A \in 2^{\Omega}$, with $inf\emptyset = \infty$. Prove that for any fixed set A, the limit always exists.

Answer 1

Hint: Suppose $\delta_1 \geq \delta_2.$ Then if $\{C_i\}_{i\in \mathbb{N}} \subseteq \mathcal{C}$ such that $A \subset \bigcup_{i=1}^\infty C_i$ and $\text{diam}(C_i) \leq \delta_2,$ then also $\text{diam}(C_i) \leq \delta_1$