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Analysis question

  1. Oct 15, 2007 #1
    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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  3. Oct 15, 2007 #2

    Dick

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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?
     
  4. Oct 15, 2007 #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?
     
    Last edited: Oct 15, 2007
  5. Oct 15, 2007 #4

    Dick

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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?
     
    Last edited: Oct 15, 2007
  6. Oct 16, 2007 #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?
     
  7. Oct 16, 2007 #6

    Dick

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    Rearrange your inequality to put a_(n+1)-a_n on one side. What's on the other side?
     
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