127A Homework 4 Solutions.pdf - MAT 127A HW 4 Joshua Sumpter Homework 4 Solutions 2.4.7 Let an be a bounded sequence a Prove that the sequence defined

127A Homework 4 Solutions.pdf - MAT 127A HW 4 Joshua...

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MAT 127A HW 4 Joshua Sumpter Homework 4 Solutions 2.4.7 Letanbe a bounded sequence.a)Prove that the sequence defined byyn= sup{ak|kn}converges. b)The limit superior ofan, or lim supan, is defined bylim supan= limynwhereynis the sequence from part a of this exercise. Provide a reasonable definition for lim infanandbriefly explain why it always exists for any bounded sequence. c)Prove that lim infanlim supanfor every bounded sequence, and give an example of a sequence forwhich the inequality is strict. d)Show that lim infan= lim supanif and only if limanexists. In this case, all three share the same value.Proof.(=)To see this, we observe that inf{ak|kn} ≤ansup{ak|kn}for alln. It follows, by squeezetheorem, that limnanexists and is equal to lim infnan= lim supnan.(=)Suppose thatana.Thenanis a bounded sequence, so the sequencesxn= inf{ak|kn}andyn= sup{ak|kn}converge.Now, for all>0, there existsNNsuch that, for allnN,|yn-a|=|sup{ak-a|kn}| ≤sup{|ak-a| |kn}<. This means that lim supnan=a. A similarstatement shows that lim infnan=a. This completes the proof.2.5.1 Give an example of each of the following, or argue that such a request is impossible.a)A sequence that has a subsequence that is bounded but contains no subsequence that converges.
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  • Spring '19
  • Mathematical analysis, Limit of a sequence, Cauchy sequence, subsequence

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