(Challenging) Let C([0,1],R) be the metric space of Example 7.1.8. Let 0 denote the zero function. Then show that the closed ball C(0,1) is not compact (even though it is closed and bounded). Hints: Construct a sequence of distinct continuous functions {fn}n=1∞ such that d(fn,0)=1 and d(fn,fk)=1 for all n=k. Show that the set {fn:n∈N}⊂C(0,1) is closed but not compact. See chapter 6 for inspiration.