In J.S. Milne's Class Field Theory notes, the following is claimed:
For finite abelian extension of local fields $L/K$,
\begin{align*} \widehat {K^\times} :=& \lim_\longleftarrow \frac{K^\times}{\text{Nm}_{L/K}L^\times } \cong \lim_\longleftarrow \frac{K^\times}{(1+ \mathfrak m^n)\langle \pi^m \rangle}\\ \cong &\lim_\longleftarrow \frac{U_K \times \pi^{\mathbb Z}}{(1+ \mathfrak m^n)\langle \pi^m \rangle} \cong \lim_\longleftarrow \frac{U_K}{1+ \mathfrak m^n} \times \lim_{\longleftarrow} \frac{\pi^{\mathbb Z}}{\langle \pi^{m} \rangle} \cong U_K \times \hat {\mathbb Z} \end{align*} ($U_K$, $\pi$, $\mathfrak m$ are respectively unit group, uniformizer, and the prime ideal of $K$)
But I don't understand how the first isomorphism arises, because I don't know if every norm group is of the form $(1+\mathfrak m^n) \langle \pi^m \rangle$.
Could anyone explain this to me, or the larger question of isomorphism $\lim_\leftarrow \frac{K^\times}{\text{Nm}_{L/K} L^\times} \cong U_K \times \hat{\mathbb Z}$?
Theorem 1.4 of Chapter I of the notes says that the norm groups in $K^{\times}$ are exactly the subgroups of finite index. The two inverse limits are equal because the subgroups $(1+\mathfrak{m}^n)\times \langle\pi^n\rangle$ are cofinal in the set of subgroups of finite index. This is explained in I, 1.10, of the notes.