American Mathematical Monthly
We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when starting with a quadratic irrational.
Northshield, S. (2011). A short proof and generalization of Lagrange’s theorem on continued fractions. American Mathematical Monthly, 118 (2), 171-175.