Let {an} be a bounded sequence. In this question, you will prove that there exists a...

Let {a_n} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence.

Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, a_m > a_n for all n > m

(a) Suppose {a_n} has infinitely many crests. Prove that the crests form a convergent subsequence.
(b) Suppose {a_n} has only finitely many crests. Let a_n1 be a term with no subsequent crests. Construct a convergent subsequence with a_n1 as the first term.

Solutions

Expert Solution

The solution is given below. In part a, the subsequence of crests is a monotonically decreasing subsequence. The infimum exists as the given sequence is bounded. This infimum is the limit. In part b, the subsequence that we get will be a monotonically increasing subsequence and the supremum will be the limit.

Colby Messinger answered 1 year ago

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