 Solution

Use the commutator convention $[x,y]=x^{-1}{y}^{-1}xy$, and let $[A,B]$ denote the subgroup generated by all such commutators with $x\in A,y\in B$. The lower central series isA central series has $[G_j,G]\le G_{j-1}$ for each $j\ge 1$. We prove the requested inclusion by induction. At $i=0$, ${\gamma }_1(G)=G_n$. If $i<n$ and ${\gamma }_{i+1}(G)\le G_{n-i}$, thenThereforeIn particular ${\gamma }_{n+1}(G)=1$. The proof uses the central-series condition directly; no equality of the intermediate terms is claimed.