 Solution

Let $I$ be a nonzero integral ideal. By the stated estimate, choose $0\ne \alpha \in I$ withSince $(\alpha )\subseteq I$, the fractional idealis integral. Its class is $[I{]}^{-1}$ andEvery 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 andFor each of the finitely many integers $m\le C$, such ideals correspond to ideals of the finite ring ${\mathcal{O}}_K/m{\mathcal{O}}_K$, of which there are only finitely many. Hence $\mathrm{Cl}(K)$ is finite.