Show that if $G$ is a finite nilpotent group, then every Sylow subgroup is normal in $G$.
I know that the normalizer of any proper subgroup of a nilpotent group contains this subgroup properly. So I think maybe I can prove the normality of the Sylow subgroup of $G$, say $P$, by showing that $N(P)=N(N(P))$, thus showing that $N(P)=G$. (Here, $N(P)$ represents the normalizer of $P$ in $G$.) But I don't know how to complete this step.
I'll appreciate your help. Many thanks.