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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

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Try monotonicity. Sorry, you're asking for the exact limit. –  Pocho la pantera Sep 23 '13 at 23:35

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(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 $x=y$.

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What's the limit in b) ? –  Martin Argerami Sep 25 '13 at 4:15

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