= Solution
Choose a nonzero <integral ideal> $I$ in the inverse of any desired <ideal class>. For the nonzero element $x\in I$ supplied by the norm bound, $J=(x)I^{-1}$ is <integral>, belongs to the desired class and satisfies
$$
N(J)=\frac{|N_{K/\mathbb Q}(x)|}{N(I)}\leq C_K.
$$
There are only finitely many <integral ideals> of bounded norm: identify $\mathcal O_K$ with $\mathbb Z^n$ as an additive group, so ideals of norm $m$ are among its <subgroups> of index $m$, of which there are finitely many. Since every <ideal class> has one of these bounded-norm representatives, \b[the <ideal class group> is finite].
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