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

Solution

(a) Example: The sequence an = (-1)ⁿ

has infinite number of ones, is bounded below by -1 and bounded above

by 1, and so is bounded. This sequence does not converge, though; since

| an- an₊₁ |= 2 for all n, this sequence fails the Cauchy criterion, and hence

diverges.

(b) no it is not possible because if an infinite sequence has limit L, then any infinte subsequence has the same limit L.

(c) No it is not possible because the sequence of real numbers is infinite sequence and if it is possible to find n consecutives ones somewhere in the sequence then it becomes the only limit of point of the sequence. Then it becomes a convergent sequence.