Question 6
Sequences $a_0,a_1,a_2,\ldots$ and $b_0,b_1,b_2,\ldots$ satisfy $a_0=2$, $b_0=1$,
$$a_{n+1}=\frac{2a_nb_n}{a_n+b_n},\qquad b_{n+1}=\sqrt{a_{n+1}b_n}.$$
Show that the two sequences converge to the same limit and determine that limit.
$$a_{n+1}=\frac{2a_nb_n}{a_n+b_n},\qquad b_{n+1}=\sqrt{a_{n+1}b_n}.$$
Show that the two sequences converge to the same limit and determine that limit.