= Solution
Let $I$ be a nonzero integral ideal. By the stated estimate, choose $0\ne\alpha\in I$ with
$$
|N_{K/\mathbb Q}(\alpha)|\leq C N(I).
$$
Since $(\alpha)\subseteq I$, the fractional ideal
$$
J=(\alpha)I^{-1}
$$
is integral. Its class is $[I]^{-1}$ and
$$
N(J)=\frac{|N_{K/\mathbb Q}(\alpha)|}{N(I)}\leq C.
$$
Every ideal class therefore has, after taking inverses, an integral representative of norm at most $C$.
There are only finitely many integral ideals of bounded norm. Indeed, if $N(J)=m$, then the finite group $\mathcal O_K/J$ has order $m$, so multiplication by $m$ kills it and
$$
m\mathcal O_K\subseteq J.
$$
For each of the finitely many integers $m\leq C$, such ideals correspond to ideals of the finite ring $\mathcal O_K/m\mathcal O_K$, of which there are only finitely many. Hence $\operatorname{Cl}(K)$ is finite.
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