I had come across a problem practicing to get better at approaching different types of problems from different field topics and this one had got me kind of stuck in what direction to go. Not so familiar with the topic, its on invariance's, so I was hoping I can get some assistance on it. The problem is the following.
An algorithm is defined as follows:
$\hspace{1.0in}$ Start: $(x_0,y_0)$ with $~0
$\hspace{1.0in}$ Step: $x_{n+1}=\dfrac{x_n+y_n}{2}, \hspace{0.4in} y_{n+1}=\sqrt{x_{n+1}y_n.}$
and the arithmetic mean-geometric mean inequality show that
$x_n
for all n. Can it be shown that the common limit, $\lim~x_n=\lim~y_n=x=y.$
Thanks.