Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 22H / c / i / Solution

Suppose first that $\Vert x_n-x_{\infty }\Vert \to 0$. The reverse triangle inequality givesso (i) implies (ii).

It also implies (iii). Given $ϵ>0$, choose $N$ so thatBy the Parseval identity for a Hilbertian basis, choose $I_0$ so that the basis tail of $x_{\infty }$ beyond $I_0$ is less than $ϵ/8$. For $n\ge N$ and $I\ge I_0$,There are only finitely many $n<N$, so increasing $I$ makes each of their tails less than $ϵ$. Thus one $I$ works for every $n$, proving (iii).