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This question already has an answer here:

Show that if $a_1>b_1>0,a_{n+1}=\sqrt{a_nb_n}$ and $b_{n+1}=\frac{a_n+b_n}{2}$, then $a_n$ and $b_n$ both converge to a common limit.

It seems a bit intuitive but I am not able to get it in writing. I tried using Cauchy's test but it isn't helping(introducing more variables). Some help please. Thanks.

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marked as duplicate by Arnaud D., Eevee Trainer, YiFan, Leucippus, Lord Shark the Unknown Mar 15 at 5:37

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One has $$a_1>b_2>b_3>b_4>\cdots>a_4>a_3>a_2>b_1.$$ So both $(a_n)$ and $(b_n)$ converge, to $A$ and $B$ say. Then $b_{n+1}=(a_n+b_n)/2\to (A+B)/2$. But $b_{n+1}\to B$.

The common limit is the arithmetic-geometric mean (AGM).

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