Let $K_0 \subset K_1 \subset ...K_n\subset...$ be a tower of normal number fields, and put $G_n = \mathrm {Gal} (K_n/K_0).$ Define epimorphisms $\pi_{mn} :G_n\longrightarrow G_{m}$ for $m < n$ such that $(G_n,\pi_{mn},\mathbb N)$ becomes a projective system,
1-How to prove that $\Gamma= \varprojlim G_n$ is topologically isomorphic to the Galois group of $K_\infty = \bigcup_n K_n/K,$ endowed with the Krull topology ?
Now we view $K_\infty/K$ as a $\mathbb Z_p-$extension of number field $K.$ Put $A_n=Cl_p(K_n)$ the $p-$Hilbert-class group of $K_n$ and denotes by $N_{mn}=N_{K_n/K_m}$ the relative norms $N_{nm}:A_n\longrightarrow A_m$ for $m<n.$
2-How to prove that $(A_n,N_{nm},\mathbb N)$ is a projective system ?