1. The problem statement, all variables and given/known data Let (M,d) be a complete metric space and define a sequence of non empty sets F1[itex]\supseteq[/itex]F2[itex]\supseteq[/itex]F3[itex]\supseteq[/itex] such that diam(Fn)->0, where diam(Fn)=sup(d(x,y),x,y[itex]\in[/itex]Fn). Show that there [itex]\bigcap[/itex]n=1∞Fn is nonempty (contains one element). 2. Relevant equations 3. The attempt at a solution We wonna use the completeness of M somehow. Let (xn) be a sequence of elements such that xn[itex]\in[/itex]Fn. Then as diam(Fn)->0 we must have for a specific N that lxn - xml < ε for all m,n>N. Thus the sequence of (xn) is a Cauchy sequence and must be convergent in M due to the assumed completeness. Denote the limit by x. We must show that x[itex]\in[/itex]Fn for all n. But I am unsure how to this. And is this even the right approach? I don't have a lot of experience with proofs.