= Solution
Choose a nonzero <integral ideal> $B$ whose class is $[A]^{-1}$; one obtains such an <ideal> by multiplying the <fractional ideal> $A^{-1}$ by a suitable nonzero <integer>. The assumed <Minkowski bound for ideal classes> gives $0\ne\beta\in B$ with
$$
|N_{K/\mathbb Q}(\beta)|\le c_KN(B).
$$
Because $(\beta)\subseteq B$ and nonzero ideals of the <ring of integers> are invertible,
$$
I_A=(\beta)B^{-1}\subseteq\mathcal O_K.
$$
It is an <integral ideal> in class $[B]^{-1}=[A]$. By multiplicativity of the <ideal norm>,
$$
\boxed{N(I_A)=\frac{|N_{K/\mathbb Q}(\beta)|}{N(B)}\le c_K.}
$$
The same argument applies if the starting $A$ is a <fractional ideal>, since its class is unchanged by clearing denominators.
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