In my lecture notes on growth of groups, we are given two definitions for a $c$-step nilpotent group $G$. It goes as follows:
A group $G$ is called $c$-step nilpotent if its lower central series, $\gamma_i(G)$ stabilizes at step $c$, i.e. $\gamma_c(G) = 1$ and $\gamma_{c-1}(G) \ne 1$. Equivalently, $G$ is $c$-step nilpotent if its upper central series, $Z_i(G)$ stabilizes at step $c$, i.e. $Z_c(G) = G$ but $Z_{c-1}(G) \ne G$.
Recall that here, the LCS is defined recursively as $\gamma_0(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G),G]$ where $[G,H]$ is the commutator of $G$ and $H$, or $[G,H] = \langle [x,y] \mid x \in G, y \in H\rangle$. The UCS is defined recursively as $Z_0(G) = 1$ and $Z_{i+1}(G) = \{ g \in G \mid [g,h] \in Z_i(G) \text{ for all } h\in G\}$.
It is not clear to me why these two definitions should be equivalent. Could anyone give some insight into why they are?