 Solution

We induct on the nilpotency class $s$. For $s=1$, the group is Abelian, so take $C_1=A$ and no $X_i$. For $s>1$, part (i) writeswith $X$ small and every $⟨B_i⟩$ of class at most $s-1$. Apply the induction hypothesis to each $B_i$. Since $B_i\subseteq A^{O(1)}$ and its approximation parameter is $K^{O(1)}$, all resulting small sets lie in $A^{O_s(1)}$, all their sizes are at mostand all resulting approximate groups lie in $A^{O_s(1)}$ and have approximation parameter $K^{O_s(1)}$. Their generated subgroups are abelian groups.

There are $O({\log }^{O(1)}(2K))$ factors at each of at most $s$ induction levels. Absorbing the resulting products of the bounds into the $O_s$ notation givesKeeping the factors in the order supplied by the induction yields the required product of the $X_i$ and $C_j$ containing $A$.