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The non-constructive version of Pataraia's theorem is called the Bourbaki-Witt fixed point theorem. I learned it from Davey and Priestley's Introduction to Lattices and Order, and Wikipedia gives the following references: Nicolas Bourbaki (1949). "Sur le théorème de Zorn". Archiv der Mathematik. 2: 434–437. doi:10.1007/bf02036949. Ernst Witt (1951). "...


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Here is a detailed outline, not entirely made rigorous. Setting $b_n \stackrel{\rm def}{=} \frac{1}{6}-Y_n$, with $b_0 = 1/6$, we have $$ b_{n+1} = \frac{4}{3}\left( \sqrt{1+\frac{3}{2} b_n - \frac{9}{2} b_n^2} - 1\right)\tag{1} $$ (I like to set things near zero.) By induction, $b_n \geq 0$ for every $n$, and a simple computation shows that $b_{n+1} - b_n \...


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