Subgroups of nilpotent groups are subnormal 
This lemma is taken from Robinson's A Course in the Theory of Groups. Why does $$\zeta_{i+1}G/\zeta_iG=\zeta(G/\zeta_iG)$$ imply that $H\zeta_iG\triangleleft H\zeta_{i+1}G$? I'll post my answer, if I find one.

First of all, given any group $G$ and a subgroup $H$ one has that
$$Z(G)\triangleleft C_G(H)\triangleleft N_G(H)$$
So that in fact $H\triangleleft H\ Z(G)$. Now consider the general case of a group $G$ with a normal subgroup $N$ and a subgroup $K$ containing $N$ such that $ K/N=Z(G/N)$, and take any $H\leq G$: then
$$HK/N=HN/N\ HK/N= HN/N\ Z(G/N)$$
thus $HN/N\triangleleft HK/N$ and by the correspondence theorem $HN\triangleleft HK$.
Setting $N=\zeta_i(G)$ the claim above holds.
