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Math Help - Convergent sequence

  1. #1
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    Convergent sequence

    Note: This is a homework problem.

    Suppose that { {a_n}} is a bounded sequence of reals such that for any n,  a_n \le \frac{a_{n-1} + a_{n+1}}{2} .

    Show that b_n = a_{n+1} - a_n is an increasing sequence and, {a_n} converges.



    I've already done the first part, that  {b_n} is an increasing sequence. For the second part, my attempt so far has been that looking at  a_2 \le \frac{a_1 + a_3}{2} , there are two possibilities:

    Case 1: It's possible that  a_3 \ge a_2 . In that case, if for some integer k,  a_{k+1} \ge a_k , then we can use that  a_{k+1} \le \frac{a_k + a_{k+2}}{2} to show that  2*a_{k+1} \le a_k + a_{k+2} \le a_{k+1} + a_{k+2} , and so  a_{k+1} \le a_{k+2} , establishing by induction that {  a_n} is also an increasing bounded sequence and then convergent.

    Case 2: It's also possible that  a_1 \ge a_2 . This is where I'm having trouble. The only way I can see to continue with this is to show that the sequence { a_n} is decreasing in this case. After several failed attempts, I'm starting to become skeptical that it actually is decreasing. Is this not going to work, and should I abandon this attempt? If so, does anyone have a suggestion for what direction to go in?
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  2. #2
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    In the second case, I assume that  a_k \ge a_{k+1} for some positive k. Then,  a_{k+1} \le \frac{a_{k+2} + a_k}{2} , implying that  2a_{k+1} \le a_{k+2} + a_k .

    I have no idea where to go after this point. It doesn't seem like I can finish this in this way by showing that the sequence is either monotonically increasing (case 1) or decreasing (case 2). Is there a better way to attempt this?

    Thanks for any help!
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  3. #3
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    Can you show that (b_n) is bounded above?
    Increasing bounded sequences converge.
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  4. #4
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    Showing that  (b_n) is bounded above follows from the fact that  (a_n) is bounded, right? I can just pick an N that's larger than  2*a_n for any n, and then  N > b_n = a_{n+1} - a_n ? So I can conclude that  (b_n} \rightarrow b , for some b. Then I'd use this to show that the  (a_n) sequence converges by taking some  \epsilon > 0 and noting that for some N, if n > N then  |b + a_{n+1} - a_n| = |b - b_n| < \epsilon .

    I don't really see how the left hand side of that last equality can be simplified to show the convergence of the  (a_n) sequence though :/

    Edit: Sorry, I'm not trying to fish for answers. I had tried looking at the limit of the  (b_n) sequence already and didn't have any luck simplifying the expression. I'm really frustrated trying to see how to evaluate limits in this way, so even a nudge in the right direction would be greatly appreciated.
    Last edited by Math Major; October 15th 2010 at 05:16 AM.
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  5. #5
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    b_n converges to 0 and a_n is a decreasing sequence. when a_n reaches its bound, b_n maxes out at 0
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