设 $A>0, x_1>0$, \[x_{n+1}=\frac{1}{2}\left(x_n+\frac{A}{x_n}\right),\ n\in\mathbb{N}_+.\] 证明: \[\lim_{n\to\infty}x_n=\sqrt{A},\] 且有 \[|x_{n+1}-\sqrt{A}|=\frac{(x_n-\sqrt{A})^2}{2x_n}\leq\frac{(x_n-\sqrt{A})^2}{2\sqrt{A}}.\]

设 $A>0, x_1>0$, \[x_{n+1}=\frac{x_n(x_n^2+3A)}{3x_n^2+A},\ n\in\mathbb{N}_+.\] 证明: $\{x_n\}$ 收敛于 $\sqrt{A}$, 且有 [|x_{n+1}-sqrt{A}|=frac{|x_n-sqrt{A}|^3}{3x_n^2+A}leq frac{|x_n-sqrt{A}|^3}{A}.]