The author gives a very simple proof for the known theorem: ``Let \(K={{\mathbb Q}}(\sqrt{m})\) be a quadratic number field with ring of integers \({\mathbb Z}_K={\mathbb Z}[\omega]\) and discriminant \(\Delta,\) where \(\omega= \sqrt{m}\) if \(m\equiv 2\) or \(3 \pmod{4}\), \(\omega={(1+\sqrt{m}) / 2}\) if \(m\equiv 1 \pmod{4}\), \(\Delta=4m\) if \(m\equiv 2\) or \(3 \pmod{4}\) and \(\Delta=m\) if \(m\equiv 1 \pmod{4}\). Let \(\mu_K\) be defined by \(\mu_K=1\) if \(\Delta=5\), \(\mu_K=\sqrt{{\Delta / 8}}\) if \(\Delta \geq 8\) and \(\mu_k=\sqrt{{-\Delta / 3}}\) if \(\Delta < 0\). Then each ideal class of \(K\) contains an integral ideal of norm \(\leq \mu_K\).'' The generalization of this result is also shown for the quadratic extension \({K / k}\) of Euclidean number fields \(k={{\mathbb Q}}(\sqrt{-n})\).