Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 19 / c

The supplied bracket gives $0<e_0=1.5-\sqrt{2}<0.1$. Part (b) ensures $x_n>\sqrt{2}>1.4$, so each error satisfiesWriting $d_n=e_n/2.8$ gives $d_{n+1}<d_n^2$ and $d_0<0.1/2.8=1/28$. After three steps the exponent has doubled three times:This rigorous error bound uses only the given bracket, not advance knowledge of the digits of $\sqrt{2}$.

The actual rational iterates are $x_1=17/12$, $x_2=577/408$, and $x_3=665857/470832\approx 1.414213562374690$. They illustrate the bound, but the proof comes from the exact error identity and the positive lower bound on the denominators.