I need to show subgroups of nilpotents groups are nilpotent.

I can get started but cannot finish.

Suppose G is a nilpotent group. Let H be a proper subgroup of G. Let K be a proper subgroup of H. I must show K is a proper subgroup of $\displaystyle N_H(K)$.

I assume $\displaystyle N_H(K)=K$. I know I must get a contradiction here but I can't figure it out.

Any hints?