Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

enter image description here

The (a) part is quite like this one These two sequences have the same limit So that I proved that they have the same limit. But I still can not find that exact limit. For (b) I have no idea.... Thanks everyone in advance for helping

share|cite|improve this question
Try monotonicity. Sorry, you're asking for the exact limit. – Pocho la pantera Sep 23 '13 at 23:35

(a) Note that $x_{n+1}y_{n+1}=x_ny_n$. Since you have already shown that $(x_n)$ and $(y_n)$ converge to a common limit $L$, we conclude $L^2=x_0y_0=ab$ and hence $L=\sqrt{ab}$. Note that $x_n,y_n>0$ for each $n\in\mathbb N_0$, since $a,b>0$.

(b) Show by induction $x_n\leq x_{n+1}$ and $y_n\geq y_{n+1}$ and $x_n\leq y_n$ which is not difficult, and deduce that $(x_n)$ and $(y_n)$ converge to $x$ and $y$, respectively. By definition of $(x_n)$ we conclude $x=\frac{x+y}{2}$ and hence $x=y$.

share|cite|improve this answer
What's the limit in b) ? – Martin Argerami Sep 25 '13 at 4:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.