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We were given those series in class:

$a_{n+1} = \frac{a_n +b_n}2$

$b_{n+1} = \sqrt{a_nb_n}$

when $a_1, b_1$ are both positive

and we were asked to prove that this limits exist and equal:

$\lim_{x\to 0}a_n = \lim_{x\to 0}b_n $

I could prove they exist but not necessarily equal through this (for every n):

$a_n > a_{n+1} > b_{n+1} > b_n$

Can someone provide a formal proof

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You already know that the limits exist, so let $\lim a_n = a$ and $\lim b_n = b$. They obviously satisfy $a = \frac{a+b}{2}$ (and $b=\sqrt{ab}$) which implies $2a = a+b$ which implies $a=b$.

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