Proving that the $[g,x]^n=e$ if $G$ is nilpotent of degree $n$

Question

This is an article from wikipedia which I saw wondering as to how to prove it.

The question is If $G$ is nilpotent of degree $n$ then $[g,x]^n=e$ for all $x \in G$, where $[g,x]=g^{-1}x^{-1}gx$.

I am only aware of the definition that $G$ is nilpotent if $Z_c(G)=G$. I have no idea as to how to prove this. I someone can give any hints it would be great.