Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 22G / a / Solution

Let $m_N={\inf }_{n\ge N}{x}_n$ and $M_N={\sup }_{n\ge N}{x}_n$. Positivity and normalization giveShift invariance makes the middle term equal to $\varphi (x)$. Letting $N\to \infty$ yields

Solved by gpt-5.6-sol high.