Give an example of each of the following, or argue that such a request is imposs

Question

Give an example of each of the following, or argue that such a request is impossible.

(a) A sequence that does not contain 0 or 1 as a term but contains subsequences converging to each of these values.
(b) A monotone sequence that diverges but has a convergent subsequence.
(c) A sequence that contains subsequences converging to every point in the infinite set {1, 1/2, 1/3, 1/4, 1/5, ...}.
(d) An unbounded sequence with a convergent subsequence.
(e) A sequence that has a subsequence that is bounded but contains no subsequences that converge.

Explanation / Answer

a) Example. Consider the sequence (an) given by an = ( 1 + 1/n if n is odd 1/n if n is even. Clearly, (1 + 1/n) ! 1 and (1/n) ! 0, so the subsequence of odd terms converges to 1 and the subsequence of even terms converges to 0, even though neither 1 nor 0 appear as terms of the sequence. b) This is impossible. To see why, suppose (an) is a monotone sequence with a convergent subsequence (ank ). In fact, for simplicity, assume (an) is increasing. Since (ank ) converges, it must be bounded, so there exists M 2 R such that ank  M for all of the ank . Now, for any term an from the original sequence, there exists k 2 N such that nk > n; then a1  an  ank  M. Since our choice of n was arbitrary, we see that a1  an  M for all n 2 {1, 2, 3, . . .}, so the sequence (an) must be bounded. A similar argument works when (an) is decreasing. Therefore, any monotone sequence with a convergent subsequence must be bounded and, therefore, convergent by the Monotone Convergence Theorem. c) Example. Consider the sequence (1, 1, 1/2, 1, 1/2, 1/3, 1, 1/2, 1/3, 1/4, 1, 1/2, 1/3, 1/4, 1/5, 1, . . .). In other words, I

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