Sequences of arithmetic and geometric mean 
I am assuming my teacher wants us to conclude they converge by being bounded and monotone, since part (i) is to show that $a_n$ is monotone increasing and $b_n$ is monotone decreasing.
I am confused though because I have never worked with a sequence defined with... another sequence.

By AM-GM inequality,
$$
b_1=\sqrt{ab}\leq\frac{a+b}{2}=a_1
$$
Assume now $b_n\leq a_n$. It follows that
$$
b_n =\sqrt{b_n^2}\leq \sqrt{a_n b_n} = b_{n+1}
$$
$$
a_n=\frac{a_n + a_n}{2}\geq \frac{a_n+b_n}{2}=a_{n+1}
$$
so $a_n$ is decreasing, $b_n$ is increasing. And (again by AM-GM)
$$
b_{n+1}=\sqrt{a_n b_n}\leq \frac{a_n + b_n}{2}=a_{n+1}
$$
so indeed $b_n\leq b_{n+1}\leq a_{n+1} \leq a_n$, as required.
