Not really homework, but I guess it's a "textbook-style question", so I'm putting it here.
Homework Statement
Suppose (X,d) is a metric space, and consider an open ball B(x,r)=\{y\in X|d(x,y)<r\}. The diameter of a subset E\subset X is defined as \text{diam\,}E=\sup\{d(x,y)|x,y\in E\}. I...