Past exam of the options II course of the University of Oxford / 2010 / E / E4 / a / ii / Solution

A subset $B$ of a metric space is bounded if it lies in some ball of finite radius: there are $x_0$ in the space and $R<\infty$ with $d(x,x_0)\le R$ for every $x\in B$. For a nonempty subset, this is equivalent to finite diameter, ${\sup }_{x,y\in B}d(x,y)<\infty$. The empty subset is bounded by convention.