Analysis question

1. Oct 15, 2007

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.

2. Oct 15, 2007

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?

3. Oct 15, 2007

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?

Last edited: Oct 15, 2007
4. Oct 15, 2007

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?

Last edited: Oct 15, 2007
5. Oct 16, 2007

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?

6. Oct 16, 2007

Dick

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