I am reading a book named 'cohomology of number fields' by Neukirch, Schmidt, Wingberg. Let $G$ be a pro-$p$-group. Suppose the $p$-cohomological dimension $cd_p G=n<\infty$. Suppose $H^n(G, \mathbb{F}_p)$ is finite where $\mathbb{F}_p$ is the field with $p$ elements with trivial $G$-action. Then in the remark of proposition 3.3.8 (page 175 of 'cohomology of number fields' (2nd ed. corrected ver 2.3. May 2020)), the authors say that $H^n(U, \mathbb{F}_p)$ is finite for all open subgroups $U$ of $G$. I don't know why. Plz anyone give me the proof of this fact.
As far as I know, if we have that $H^i(G, \mathbb{F}_p)$ is finite for all $0<i\le n$, then $H^i(U, M)$ is finite for all $i$, all open subgroups $U$, and all $p$-primary finite Galois modules $M$ by usual argument of Hochschild-Serre spectral sequence. But I don't know how to control the finiteness only when we know the finiteness of $H^n(G, \mathbb{F}_p)$.