 Solution

The singular values discarded by the threshold in (i) satisfy ${\lambda }^2<c/2$, so their contribution to the fourth moment is at most $(c/2)S_2\le cn/2$. The selected singular values consequently satisfyBecause ${\lambda }_p\le 1$, this implies $m={\sum }_p{n}_p\ge cn/2$. On the other hand, ${\lambda }_p^2\ge c/2$, and henceThusThe lower bound also shows that the selected list is nonempty. Both endpoints are inclusive, as required.