With $x_{n+1}=\sqrt{x_ny_n}$, and $y_{n+1}={x_n+y_n \over 2}$, $n\geq 2$. Show that $\lim _{n\rightarrow \infty}x_n=\lim_{n\rightarrow\infty} y_n$.
prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit