Every unbounded sequence has a monotone subsequence that diverges either to+∞or to-∞.
Limit Superior and Limit InferiorDefinition 7.
Let{sn}be a bounded sequence. A numberαis asubsequential limitof{sn}if there is a subsequence{snk}of{sn}such thatsnk→α.ExamplesLet{sn}be a bounded sequence. LetS={α:αis a subsequential limit of{sn}.Then:1.S=∅.2.Sis a bounded set.
Let{sn}be a bounded sequence and letSbe its set of subsequential limits. Thelimit superiorof{sn}(denoted by lim supsn)islim supsn=supS.Thelimit inferiorof{sn}(denoted by lim infsn)islim infsn=infS.ExamplesClearly, lim infsn≤lim supsn.
Let{sn}be a bounded sequence.{sn}oscillatesiflim infsn<lim supsn.
Exercises 2.4
1. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counter-example.(a) A sequence{sn}converges tosif and only if every subsequence of{sn}converges tos.
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(b) Every bounded sequence is convergent.(c) Let{sn}be a bounded sequence. If{sn}oscillates, then the setSof subsequentiallimits of{sn}has at least two points.(d) Every sequence has a convergent subsequence.(e){sn}converges tosif and only iflim infsn= lim supsn=s.2. Prove or give a counterexample.(a) Every oscillating sequence has a convergent subsequence.(b) Every oscillating sequence diverges.(c) Every divergent sequence oscillates.(d) Every bounded sequence has a Cauchy subsequence.(e) Every monotone sequence has a bounded subsequence.
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- Fall '08
- Staff
- Real Numbers, Integers, Limit of a sequence, Sn