Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 22H / d / Solution

Given $\epsilon >0$, choose $N$ with ${\sum }_{i>N}|⟨a,e_i⟩{|}^2<{\epsilon }^2$. Every cube point has uniformly bounded tail by this sum, while its first $N$ coordinates lie in a compact finite-dimensional box. A finite net for that box is therefore an $\epsilon$-net for the cube. The coordinate inequalities define a closed set, so the cube is complete and totally bounded, hence compact.