Could someone please check my work

2. Find an infinite collection {S, : n E N} of compact sets in IR such that () Sn is not compact. n =1 By Theorem 3.5.5 (Heine Borel), the set USn is not compact if it is not both closed and n =1 bounded. WTS | Sn is not compact by showing that this family of set unions, Sn is unbounded. n=1 Let Sn = man + 1 , which is a closed and bounded set interval. U Sn = [1, 2] U [ 2 , 3] U [ , 4 ] U .. U [m, n + 1] - (0,00 ) n =1 Since (0, oo) is unbounded, ( Sn is not compact by Theorem 3.5.5 (Heine Borel). n= 1 OO Therefore, \\ Sn is not compact when Sn = n =1