Let $a$ be a positive real number, and let $x_1 > \sqrt a$. Define the sequence $\{x_n\}$ where $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$
I would like to show that $x_n > \sqrt a$ for every $n$, but I am unable to show this. Can anyone give me any ideas?