Solution
= Solution
Given $ε>0$, choose $N$ with $\sum_{i>N}|\langle a,e_i\rangle|^2<ε^2$. 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 $ε$-net for the cube. The coordinate inequalities define a closed set, so the cube is complete and totally bounded, hence <compact set>[compact].