Could someone please check my work?

Show That compactness is necessary in This corollary: ( Nesred Intervals Theorem) Ler F= EAN: nEAN] be a family of closed bounded intervals in IR Such That A SA for all newN. Then NA * . That is , Find a family of intervals EAn: nEIN] with Any, SAn for alln, MA, ED, and such That The sers An are all closed. n=l Ler An = [n, 00). This is an unbounded closed interval. -> A , = [1, 00 ) A, CA2 SA, A2 = [2, 60 ) O- AL A3 = [3, 00 ) A2 . .. . A 3 A = [n, 00 ) 2 W A, 2 A 2 2 A 3 2 ... An 00 * nA = @ because [A 3 was not bounded n = 1