Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / c / Solution

Changing $X_k$ while keeping all other coordinates fixed changes every candidate linear form by at mostThe maximum of finitely many functions obeys the same bound, so part b applies with $c_k=2{\max }_{\ell }|A_{k,\ell }|$. Therefore