By Krantz S.G.

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1). But this last is < + = 1. We have proved that 2 < 1, a clear contradiction. So the sequence {bj } has no limit. 5 Given any sequence, it either converges or it diverges. There is no in-between status, and no undecided status. We begin with a few intuitively appealing properties of convergent sequences which will be needed later. First, a definition. 6 A sequence aj is said to be bounded if there is a number M > 0 such that |aj | ≤ M for every j. 7 Let {aj } be a convergent sequence. Then we have: • The limit of the sequence is unique.

And {bj } is bounded above. 15. * 9. Give an example of a sequence with infinitely many distinct subsequences that converge to π. 10. Give an example of a sequence which does not converge, but which has infinitely many different subsequences that do converge. 11. Prove that a sequence {aj } converges if and only if every subsequence has a subsequence that converges. 3. 3 31 Lim sup and Lim inf Preliminary Remarks While our interest in sequences is in their limits, it is a fact that most sequences do not have a limit.

Let aj+1 λ = lim sup < 1. aj j→∞ Select a real number µ such that λ < µ < 1. By the definition of lim sup, there is an N so large that, if j > N , then aj+1 < µ. aj This may be rewritten as |aj+1 | < µ · |aj | , j≥N. Thus (much as in the proof of the Root Test) we have for k ≥ 0 that |aN+k | ≤ µ · |aN+k−1 | ≤ µ · µ · |aN+k−2 | ≤ · · · ≤ µk · |aN | .