 Solution

Choose a fixed $\alpha <1/2$. On the stated high-probability event, strictly more than half of the block means satisfy, simultaneously for every $\Vert v{\Vert }_2\le 1$,The median of a collection with a strict majority in an interval lies in that interval. Hence the same bound holds for $|{\mathrm{MOM}}_k(Xv)-{\mu }_0^{T}v|$ uniformly in $v$. Part b then giveswith probability at least $1-e^{-k/c_{\alpha }}$. Renaming the constant proves the claim.