I have attached a picture of the question. How would you go about showing this? Thanks.
I work out that if $\displaystyle x_n > \frac{a}{x_n}$,
$\displaystyle x_n + \frac{a}{x_n} > \frac{2a}{x_n}$
$\displaystyle 0.5\left(x_n + \frac{a}{x_n}\right) > \frac{a}{x_n}$
$\displaystyle x_{n+1} > \frac{a}{x_n}$.
As long as $\displaystyle x_n > \frac{a}{x_n}$, then the function will be decreasing, but I can't find a way to show that statement inductively.