Let {an} be a bounded sequence. In this question,
you will prove that there exists a...
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Let {an} be a bounded sequence. In this question,you will prove that there exists a convergent subsequence.
Define a crest of the sequence to be aterm am that is greater than all subsequent terms. That is,am > an for all n > m
(a) Suppose {an} has infinitely many crests. Provethat the crests form a convergent subsequence.
(b) Suppose {an} has only finitely many crests. Letan1 be a term with no subsequent crests. Construct aconvergent subsequence with an1 as the first term.
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The solution is given below In part a the subsequence
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