Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 3 / b / Solution

Apply part a to each of the at most $n(n-1)/2$ nonzero differences $u_i-u_j$. For $w_i=A{u}_i/\sqrt{d}$, the union bound makes the probability of any failure at mostThe assumed inequality $d>272\log (n/\sqrt{\epsilon })/t^2$ makes this smaller than $\epsilon$. Hence, simultaneously for every distinct pair,with probability at least $1-\epsilon$, which is the finite-set Johnson–Lindenstrauss lemma.