I was reading about the Nested sphere theorem and a thought occurred. if you have a sequence of decreasing closed sets whose diameter goes to zero in the limit, we can show that the intersection of all these sets is a single point. my idea was to show this using nested sphere theorem if we can say that for any closed set we can find a smallest closed sphere containing this set as well as a biggest sphere contained in this set. that way we can trap our original sequence between two sequences of decreasing spheres. Is this if fact possible? if so/not, why? cheers!