 Solution

For $u=0$ the assertion is immediate under the natural zero-vector convention. For $u\ne 0$, the rows $a_r$ of $A$ are independent and $a_r^{T}u/\Vert u{\Vert }_2$ is a centered unit-variance sub-Gaussian random variable. Its centered square is sub-exponential. The corresponding Bernstein estimate, in the explicit Rademacher Johnson–Lindenstrauss transform form, isSince the sum in the event is $\Vert Au{\Vert }_2^2/\Vert u{\Vert }_2^2$, this is the claimed inequality.