The purpose of this problem is help you come to terms with the fact that things that are true

in a finite setting may not extend to be true in an infinite setting. For example, infinite sets

need not have a smallest element, e$.$g$.$ A $=\{1,1/2,1/3,1/4,...\}.$

For each part below, suppose that A$1,$ A$2,$ A$3,...$ form an infinite sequence of nested sets

such that