upper central series, taking intersection with subgroup ($\zeta_i G \cap H \leq \zeta_i H$)

upper central series, taking intersection with subgroup ($\zeta_i G \cap H
\leq \zeta_i H$)

Let $\zeta_i G$ be the upper central series of $G$. Show that for 'any'
subgroup $H$, we have
$$\zeta_i G \cap H \leq \zeta_i H$$
where $\zeta_i H$ is the upper central series of $H$.



I tried induction. The case $n=0$ was easy. Then things get messy.
My question stems from a small step in a proof. There they also presuppose
that $G$ is nilpotent of class $\geq 2$ and $H= \left < a, [G,G] \right >$
with $a \in G$.
However, I do not suspect these conditions are necessary and that the
general case also holds.