It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, $g_{n+1} = \sqrt{a_ng_n}$ are both convergent and that $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} g_n$. Let's call this limit $\operatorname{agm}(\alpha, \beta)$. I am convinced that the number $\operatorname{agm}(1,2)$ is irrational, but I don't know how to prove it.

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