Proof of Limit of Bounded Sequence: 2an <= an+1 + an-1

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Homework Help Overview

The discussion revolves around proving a limit related to a bounded sequence defined by the inequality 2an ≤ an+1 + an−1. The original poster is seeking guidance on how to approach this proof in the context of elementary analysis.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the given inequality and its relationship to the behavior of the sequence, particularly whether an+1 - an is bounded and increasing. Questions are raised about the necessity of the limit being zero and how to demonstrate boundedness.

Discussion Status

The conversation is active, with participants offering hints and exploring various interpretations of the inequality. Some guidance has been provided regarding the properties of the sequence, but no consensus has been reached on the necessity of the limit being zero.

Contextual Notes

Participants are working under the constraints of a homework problem, which may limit the information available for discussion. The original poster expresses uncertainty about the implications of boundedness and the behavior of the sequence.

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Let (an) be a bounded sequence. If 2an <= an+1 + an−1, prove that
limit n to infinity of (an+1 − an) = 0.


This question is on my exam review sheet for an elementary analysis class and I'm not really sure where to start. Could someone just give me a hint or something? I'm pretty sure an is decreasing and therefore convergent since it's bounded. But I'm not sure how to conclude this or arrive at the final conclusion.
 
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Use your inequality to show a_(n+1)-a_n is an increasing sequence and is bounded. So it has a limit. Can the limit be nonzero?
 
Ok, that makes sense but what makes the limit have to be zero? I guess I don't know how to answer your question. Why can't it be non-zero?

If an is bounded, does that mean an+1 - an is bounded also? How do I show it's bounded?
 
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If a_n is bounded, then A<=a_n,a_(n+1)<=B for some numbers A and B. Can you show a_(n+1)-a_n is bounded? If a_(n+1)-a_n approaches a nonzero limit, can a_n be bounded?
 
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OK. I think I figured it out mostly, but I still don't see how the given inequality shows that (an+1 − an) is an increasing sequence. It holds when I take examples, but why for an arbitrary bounded one?
 
Rearrange your inequality to put a_(n+1)-a_n on one side. What's on the other side?
 

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