= Solution
The assumed inclusion and the class-field norm-index theorem give
$$
C_{\mathbb Q}(m\infty)
\subseteq N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}
\subseteq C_{\mathbb Q}.
$$
The outer subgroup has index $\varphi(m)$ by part (d), while
$$
[C_{\mathbb Q}:N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}]
=[\mathbb Q(\zeta_m):\mathbb Q]
=\varphi(m).
$$
Two nested subgroups of the same finite index are equal. Thus
$$
\boxed{C_{\mathbb Q}(m\infty)
=N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}.}
$$
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