Why does Brent/Salamin algorithm double the digits of $\pi$ with each iteration? The Brent-Salamin-Formula uses the arithmetic-geometric mean to calculate $\pi$.
There are many sophisticated proofs proving very sharp error bounds, for example in Salamin's 1976 paper or in this 2018 paper by Richard Brent, equation (20), p. 9.
Is there an easy way to prove the (much weaker) error bound $|\pi-p_{n+1}|<|\pi-p_n|^2$?
A partial solution proving the quadratic convergence of the AGM is this:
\begin{align*}
    d_{n+1}&=c_{n+1}^2 =a_{n+1}^2-b_{n+1}^2\\
    &= \left(\frac{a_n+b_n}{2}\right)^2-a_n\cdot b_n\\
    &= \frac{a_n^2+2a_nb_n+b_n^2-4a_nb_n}{4}\\
    &= \left(\frac{a_n-b_n}{2}\right)^2\\
    &=\frac{(a_n-b_n)^2\cdot(a_n+b_n)^2}{4\cdot(a_n+b_n)^2}\\
    &= \frac{(a_n^2-b_n^2)^2}{4\cdot(a_n+b_n)^2}\\
    &< \frac{\left(c_n^2\right)^2}{4\cdot\left(1/\sqrt{2}+1/\sqrt{2}\right)^2}\\
&=\frac{1}{8}\cdot d_n^2
\end{align*}

An easy proof of the quadratic convergence can be found on p. 13-14 of this preprint.
