How to show that the class number is bounded by Minkowski's bound?

Question

Let $K$ be a number field and let $M$ be the Minkowski's bound. We can say that each ideal class in the ideal class group $Cl(K)$ has an ideal $I$ with $N(I) < M$. By this fact, how can we say that $h_K < M$? I am not sure that if any two integral ideals has the same norm, they belong the same class or not. If it is, then we are done.