Discrete family of compact subsets in a metric space A family $\{A_i: i \in A_i\}$  of subsets of a topological space $X$ is said to be discrete if each $x \in X$ has a neighborhood meeting at the most one $A_i$.
Suppose $(X,d)$ is a metric space and $\{A_n: n \in \mathbb{N}\}$ a discrete family of compact subsets of $X$. How can we show that there exists a sequence $\varepsilon_{n}$ of positive number which converges to $0$ and the family $\{B(A_n,\varepsilon_{n}): n \in \mathbb{N}\}$ is a discrete family. Where $B(A_n,\varepsilon_{n})$ denotes the union of all balls centered at the points of $A_n$ and radius $\varepsilon_n$. 

This seems false:
In $\mathbb R$, choose the family $A_1=[-1,0]$ and $A_n=[2^{-2n-2},2^{-2n-1}]$ for $n\geq2$.  No matter what $\varepsilon_1>0$ we choose, there will be some $x\in B(A_1,\varepsilon_1)$ which is also in $A_n$ for some $n$.
