Need help proving any subgroup and quotient of a nilpotent group is nilpotent? $G$ is the direct product of its Sylow subgroups $P_i$. Then if $H \le G$, $P_i \cap H \le H$. I'm stuck now on how to proceede to find a terminating central series for $H$.
Also, I know $P_i H / H \le G/H$ so I'm assuming proving the above will prove that the quotient is nilpotent too?

Let $H$ be a subgroup of the nilpotent group $G$. Pick a central series
$$
   \langle1\rangle = N_0\subseteq N_1\subseteq\cdots\subseteq N_r=G.
$$
Then
$$
    \langle1\rangle=H\cap N_0\subseteq H\cap N_1\subseteq\cdots\subseteq H\cap N_r=H
$$
is a normal series in $H$ because $H\cap N_i\triangleleft H$. Moreover, for every $y\in H$ and $z\in H\cap N_i$ we have
$$
    \text{ad}_y(z)=yzy^{-1}z^{-1}\in H\cap N_{i-1}.
$$
Thus, $\text{ad}_y(H\cap N_i)\subseteq H\cap N_{i-1}$, i.e., the series for $H$ is central.
