Exercise 224 Give an example of each or state that the reque

Exercise 2.2.4. Give an example of each or state that the request is impossible. For any that are impossible, give a compelling argument for why that is the case (a) A sequence with an infinite number of ones that does not converge to one. (a) A sequence with an infnite number of ones that does not converge to one

Solution

(a) a_n = {(-1)ⁿ | when n belong to N}.

(b) The request is impossible to exist because, if an infinite sequence has a limit L, then any infinite subsequence has the same limit.

(c) a_n = {1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,...} is an example of such divergent sequence.