Proving that a rational sequence is monotonic decreasing 
Problem 4. Let $a>0$ be a real number and consider the sequence $(x_n)$ given by $$a_{n+1} = \frac{x_n^2 + a}{2x_n}$$

(original scan of problem)
I want to show that this sequence is monotonic decreasing for all $n>2$.
is induction the way to go for this, or something else?

Note that
$$x_{n}-x_{n+1}=x_n-\frac{x_n^2+a}{2x_n}=\frac{x_n^2-a}{2x_n}.$$
So we want to show that $x_n^2\ge a$. This need not be true at $n=1$, but it is true after that. For if $n\gt 1$ then 
$$x_n=\frac{x_{n-1}^2+a}{2x_{n-1}}$$ and therefore 
$$x_n^2-a=\left(\frac{x_{n-1}^2+a}{2x_{n-1}}\right)^2-a=\frac{(x_{n-1}^2-a)^2}{4x_{n-1}^2} $$
