= Solution
Start with any generator $I=(\beta)$ and use the decomposition from part (d). For each $i$, choose an integer $n_i$ such that
$$
-\frac12\leq\lambda_i+n_i\leq\frac12,
$$
and put
$$
u=u_1^{n_1}\cdots u_m^{n_m},
\qquad
\alpha=\beta u.
$$
Since $u$ is a unit, $(\alpha)=(\beta)=I$. Moreover,
$$
\operatorname{Log}(\alpha)
=te+h,
\qquad
h=\sum_{i=1}^m(\lambda_i+n_i)\operatorname{Log}(u_i).
$$
The vector $h$ lies in the fixed compact parallelepiped
$$
P=\left\{\sum_i c_i\operatorname{Log}(u_i):|c_i|\leq\frac12\right\}.
$$
Hence there is a constant $M_K$ such that every real coordinate of every $h\in P$ is at most $M_K$, and every complex coordinate is at most $2M_K$.
For a real embedding $\sigma_i$,
$$
\log|\sigma_i(\alpha)|=t+h_i
\leq\frac{\log N(I)}{[K:\mathbb Q]}+M_K.
$$
For a chosen complex embedding $\tau_j$, the weighted coordinate gives
$$
2\log|\tau_j(\alpha)|=2t+h_{r+j}
\leq2\frac{\log N(I)}{[K:\mathbb Q]}+2M_K.
$$
The conjugate embedding has the same modulus. Exponentiating and taking any $C>e^{M_K}$ proves the <archimedean balancing of a principal ideal generator>:
$$
\boxed{
|\sigma(\alpha)|<C N(I)^{1/[K:\mathbb Q]}
}
$$
for every embedding $\sigma:K\to\mathbb C$, with $C$ depending only on $K$.
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