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|k≥n}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 infan≤lim 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|k≥n} ≤an≤sup{ak|k≥n}for alln. It follows, by squeezetheorem, that limnanexists and is equal to lim infnan= lim supnan.(⇐=)Suppose thatan→a.Thenanis a bounded sequence, so the sequencesxn= inf{ak|k≥n}andyn= sup{ak|k≥n}converge.Now, for all>0, there existsN∈Nsuch that, for alln≥N,|yn-a|=|sup{ak-a|k≥n}| ≤sup{|ak-a| |k≥n}<. 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