Definition: A delta cover of a bounded set $F\subseteq \mathbb{R}^n$ is a collection of sets, each with diameter at most $\delta$ such that their union forms a cover of $F$.
Let $U$ be a set in a $\delta-cover$ (where $\delta< \frac{1}{2}$) of $F$ $=$ $\{$ 0,1,$\frac{1}{2},\frac{1}{3}$....$\}$. Let $k \in \mathbb{N}$ be such that $\frac{1}{k(k+1)}$ $\leq$ $\delta$ $<$ $\frac{1}{k(k-1)}$. Show that $U$ can cover at most one point of $\{$ $1$ , $\frac{1}{2}$,..., $\frac{1}{k}$ $\}$
I have no idea how to approach this problem, but I feel as though the pigeon hole principle should be; however, I am not sure. May someone please explain?
Hint: The minimum distance between elements of $\{0,1,1/2,1/3,\dots,1/k\}$ is $1/(k(k-1))$. Now use the triangle inequality.