Could someone please check my work

Show that compactness is necessary in Corollary 3.5.8. That is, find a family of intervals { An : n E N} with Anti C An for all n, ( An = 0, and such that n = 1 2. The sets An are all bounded. WTS An = Al n A2 n Ag n... n An = 0 when An is bounded but not n =1 closed and An+1 C An for all n. Let An 1 , 1 n , which a bounded but not closed set interval. ( - 27 2 ) - 1- 20n ( - 2 2)~ ( - 3. 3) n... ( - mim ) = 1 n = 1 Consequently, A1 2 A2 2 A3 2 ... 2 An since ( - 1 , 1) 2 ( - 2 , 2 ) = ( - 3 3 ) 2 ..2 ( - 7 7). Hence, An+1 C An Or more clearly, An 2 An+1 . Therefore, compactness is necessary in Corollary 3.5.8. When An is bounded but not closed, the corollary does not hold