|
Response Details:
If L is the limit of the subsequence, then L must be an integer. If it weren't, then for E < (distance from L to the nearest integer), there would be no elements of the whole sequence within a distance E of L. Thus the subsequence could not converge to L.
If you let E = 1/2, and let L be the limit of the subsequence, then for some N, you have | x_n - L | < 1/2 for every n < N, such that x_n is in the subsequence. But the x_n are all integers, and the only integer within a distance of 1/2 from L is L itself. So all these x_n are equal to L. So, the subsequence is eventually constant.
|