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.

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 2 years ago

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Let sequence an be a bounded sequence and let E be the set of subsequential limits...

Let sequence an be a bounded sequence and let E be the set of subsequential limits of an. prove that E is bounded and contains both sup E and inf E

Prove that if a sequence is bounded, then limsup sn is a real number.

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Let A be a subset of all Real Numbers. Prove that A is closed and bounded...

Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...

Question

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

Solutions

Expert Solution

Related Solutions

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let sequence an be a bounded sequence and let E be the set of subsequential limits...

Prove that if a sequence is bounded, then limsup sn is a real number.

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Let A be a subset of all Real Numbers. Prove that A is closed and bounded...

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove...

Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an}...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Prove: Let A be an mxm nonnegative definite matrix with rank(A)=r Then there exists an mxr...