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

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In summary, to prove that the limit of (an+1 − an) is 0 for a bounded sequence an, it is necessary to show that (an+1 − an) is an increasing sequence and is bounded. This can be done by using the given inequality and rearranging it so that a_(n+1)-a_n is on one side and a constant is on the other side. If a_(n+1)-a_n approaches a nonzero limit, then a_n cannot be bounded, leading to the conclusion that the limit must be 0.
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PvtBillPilgrim3
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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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  • #2
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?
 
  • #3
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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  • #4
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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  • #5
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?
 
  • #6
Rearrange your inequality to put a_(n+1)-a_n on one side. What's on the other side?
 

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

What is a bounded sequence?

A bounded sequence is a sequence of numbers in which there is a finite number that serves as an upper or lower bound for all the elements in the sequence. In other words, the numbers in the sequence do not exceed a certain value.

How is the limit of a bounded sequence calculated?

The limit of a bounded sequence can be calculated by finding the value that all the terms in the sequence approach as the sequence becomes infinitely long. This is often denoted as limn→∞an, where n represents the number of terms in the sequence.

What does the inequality 2an ≤ an+1 + an-1 mean in this context?

This inequality means that the value of each term in the sequence is less than or equal to the sum of the two adjacent terms. This is known as the Cauchy criterion, and it is commonly used to prove the existence of a limit in a bounded sequence.

How does the proof of the limit of a bounded sequence using the Cauchy criterion work?

The proof of the limit of a bounded sequence using the Cauchy criterion involves showing that as the number of terms in the sequence increases, the difference between any two consecutive terms becomes smaller and smaller. This ensures that the terms in the sequence are approaching a specific value, which is the limit of the sequence.

Are there any other methods for proving the limit of a bounded sequence?

Yes, there are other methods for proving the limit of a bounded sequence, such as the Monotone Convergence Theorem and the Bolzano-Weierstrass Theorem. However, the Cauchy criterion is one of the most commonly used methods for proving the limit of a bounded sequence.

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