1.4.5 Let S = { 0 , 1 , 2 , . . .} be the set of all integers. For A c S, let f n ( A ) be the number of elements in the intersection A n { 0 , l , . . . n}. Let A be the class of all sets A for which the limit q (A) = lim -fn(A)

exists. Show that A is not a field. [Hint:L et A1 = { 1,3,5,. . .} and A2 ={ all odd integers between 22n and 22n+1 and all even integers between 22n+1 and 22n+2 for n = 0, 1, . . . }. Show that both A1 and A2 are in A but A1 n A2 $ A.]