Let $E$ be a metric space. Assume $\exists x,y \in E$ and $\delta > 0$ such that $\forall n\in \mathbb N$ and $x_1,x_2,\cdots, x_n\in E$ we have $\max[ d(x,x_1),d(x_1,x_2),d(x_2,x_3),\cdots,d(x_{n-1},x_n),d(x_n,y)$] $\geq \delta $. Show that $E$ is not connected.
I'm a bit stuck on this proof. I started off by trying to prove by contradiction by creating some subset of $E$ called $A$ and fixing to a point and creating a ball whose radius is less than the $\min\{\}$ of all those distances. Following that, the complement of $A$ would include all other points. Hence, $E$ is the union of $A$ and complement of $A$. However, I feel I need to utilize the fact the $\max$ is greater than $\delta$.
