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Proving the existence of the arithmetic-geometric mean

  1. Jul 25, 2014 #1
    1. The problem statement, all variables and given/known data
    [itex][/itex]
    Given:
    1. [itex]a_{1} < b_{1}[/itex]
    2. [itex]a_{n} = \sqrt{a_{n-1}b_{n-1}}[/itex]
    3. [itex]b_{n} = \frac{a_{n-1} + B_{n-1}}{2}[/itex]
    4. The sequences [itex]a_{n}[/itex] and [itex]b_{n}[/itex] are convergent.

    Prove: The sequences [itex]a_{n}[/itex] and [itex]b_{n}[/itex] have the same limit.

    3. The attempt at a solution
    Assume by contradiction that [itex]\lim_{n\to\infty}a_{n} = l_{a}[/itex] and [itex]\lim_{n\to\infty}b_{n} = l_{b}[/itex] and [itex]l_{a} \neq l_{b}[/itex].

    Because [itex]b_{n} = \frac{a_{n-1} + B_{n-1}}{2}[/itex], we can write [itex]\lim_{n\to\infty}b_{n} = \lim_{n\to\infty}\frac{a_{n-1} + B_{n-1}}{2}[/itex].

    From arithmetic of limits, we have [itex]\lim_{n\to\infty}\frac{a_{n-1} + B_{n-1}}{2} = \frac{1}{2} (\lim_{n\to\infty}a_{n-1} + \lim_{n\to\infty}B_{n-1})[/itex].

    For all [itex]n \ge 2[/itex], the sequences [itex]a_{n}[/itex] and [itex]a_{n-1}[/itex] are identical. Thus their limit is identical, [itex]l_{a}[/itex].

    The same is true of the sequences [itex]b_{n}[/itex] and [itex]b_{n-1}[/itex]. Thus their limit is identical too, [itex]l_{b}[/itex].

    Thus, [itex]\frac{1}{2} (\lim_{n\to\infty}a_{n-1} + \lim_{n\to\infty}B_{n-1}) = \frac{1}{2} (l_{a} + l_{b})[/itex].

    We assumed, by way of contradiction, that [itex]l_{a} \neq l_{b}[/itex], that is either [itex]l_{a} < l_{b}[/itex] or [itex]l_{a} > l_{b}[/itex].

    Assuming [itex]l_{a} < l_{b}[/itex], we get [itex]\lim_{n\to\infty}b_{n} = \frac{1}{2} (l_{a} + l_{b}) < \frac{2l_{b}}{2} = l_{b}[/itex].

    Assuming [itex]l_{a} > l_{b}[/itex], we get [itex]\lim_{n\to\infty}b_{n} = \frac{1}{2} (l_{a} + l_{b}) > \frac{2l_{b}}{2} = l_{b}[/itex].

    Thus, if [itex]l_{a} \neq l_{b}[/itex], then [itex]\lim_{n\to\infty}b_{n} \neq l_{b}[/itex]. Contradiction.

    Therefore, [itex]l_{a} = l_{b}[/itex].

    ---

    Here's my question. I wasn't sure that I didn't make some logical leap along the way. Is my attempt correct? Did I manage to prove what I set out to prove? Thanks!
     
  2. jcsd
  3. Jul 25, 2014 #2
    It looks fine. I do have a couple of remarks on style though. First, call the limits ##a## and ##b##. Second, try to see if you can get a direct proof. Proofs by contradiction are perfectly fine, but when there is a very easy way to turn a proof by contradiction into a direct proof (and there is here), I've always been told that the direct proof is preferable. In this case, when you use my tips with the key part of your argument, you get $$b=\lim b_n=\lim\frac{1}{2}(a_n+b_n)=\frac{1}{2}(a+b)$$ and the desired result is a couple steps of elementary algebra away.

    Again, though, your version works. It's just a little messier.
     
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