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

[PvtBillPilgrim3]

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.

 

[Dick]

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?

 

[PvtBillPilgrim3]

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?

 

[Dick]

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?

 

[PvtBillPilgrim3]

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?

 

[Dick]

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