We have 2 definition of Nilpotent group.I am trying to prove the equivalence.
A central series for $G$ is a normal series $1=G_0\unlhd G_1\unlhd\dots\unlhd G_r=G$ such that $G_i/G_{i−1}=Z(G/G_{i−1})$ for every $i=1,\dots, r$. An arbitrary group $G$ is said to be nilpotent if it has a central series
The descending central series of a group G is $G=F_1(G)\supseteq F_2(G)\supseteq \dots$, where $F_{i+1}(G)=[F_i(G),G]$. A group $G$ is called nilpotent if the lower central series reaches $\{1\}$; that is, if $F_n(G)=\{1\}$ for some $n$.
I have proved that Ist defn will imply second one as follows suppose $G_c=G$, then I prove that $F_k(G)\le G_{c-k}(G)$ by induction on $k$. For the base case $k=0$ both are $G.$ In the induction hypothesis I used one easy exercise from Hungerford exercise 3(d) pg no 106. So this portion I did in my own but I am completely stuck for the other part, i.e assuming some $F_n=\{1\}$ I can't show $G_n=G$.
Please help me. Thank you.