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

Let $S$ be the left shift and defineShift invariance gives $\varphi (A_{p}x)=\varphi (x)$. The hypothesis says $A_{p}x$ converges uniformly in $n$, hence in the ${\ell }^{\infty }$ norm, to the constant sequence $y1$. By continuity,

Solved by gpt-5.6-sol high.