Let ${a_n}$ be a monotonic decreasing sequence.
and Let ${b_n}$ be a monotonic increasing sequence.
Its given that for each $n:$
$$b_{n+1}=\sqrt{a_nb_n}$$
Prove that both sequences converge to the same limit.
by doing some algebra, using the fact that: $a_{n+1}\le a_n$ and $b_n \le b_{n+1}$ I got that the following inequality:
$b_n \le b_{n+1} \le a_{n+1} \le a_n$.
now in order to prove that $a_n$ and $b_n$ converge to the same limit I thought about using the Lemma of Cantor. but there is one thing else that I need to prove first, which is that: $\lim \limits_{n \to \infty}$ $(a_n – b_n)=0$.
do you guys have any hints for how to do that?
Best Answer
Hint: $$a_n=\frac{b_{n+1}^2}{b_n}$$
Since you know $b_n$ converges, what is the limit of the right side?