subgroups complementing terms of the lower exponent-p central series (Huppert-Blackburn Th. VIII.1.7b)) I am trying to understand the proof of Theorem VIII.1.7b) in Huppert-Blackburn "Finite Groups. II". It deals with what is elsewhere called the lower exponent-$p$ series of a group $G$, defined as:
$$\lambda_1(G)=G,\qquad \lambda_{n+1}(G)=[\lambda_{n}(G),G](\lambda_n(G))^p.$$
Theorem VIII.1.5b) says that
If $H\le G$, and $H\lambda_2(G)=G$, then $H\lambda_n(G)=G$ for all $n\ge1$.
The proof by induction is very short but I don't understand it:

Suppose that by induction hypothesis we have $G=H\lambda_n(G)$. Then:
$$\lambda_n(G)=[\lambda_{n-1}(G),G]\lambda_{n-1}(G)^p
\le [H\lambda_n(G),H\lambda_n(G)](H\lambda_n(G))^p
\le H\lambda_{n+1}(G),$$
from where we conclude that $G=H\lambda_n(G)\le H\lambda_{n+1}(G)$, and we are done.

I do not understand the inequality $[H\lambda_n(G),H\lambda_n(G)](H\lambda_n(G))^p
\le H\lambda_{n+1}(G)$ above. There is something very obvious to the authors used, but all my fiddling with commutators did not yield the required form on the right-hand side. Any hints?

To save typing, let $A=\lambda_n(G)$ and $B = \lambda_{n+1}(G)$.
Then (since $A \unlhd G$), we have
$$[HA,HA] = [H,H][H,A][A,A] \le H[G,A] \le HB,$$
and $$(HA)^p \le H^pA^p[H,A] \le HA^pB \le HB.$$
