 Every Euclidean domain is a principal ideal domain

Let $R$ be a Euclidean domain with Euclidean function $\nu$, and let $I$ be a nonzero ideal. Choose $b\in I\setminus \{0\}$ minimizing the nonnegative integer $\nu (b)$. For $a\in I$, divide $a=qb+r$ with $r=0$ or $\nu (r)<\nu (b)$. Since $r=a-qb\in I$, the latter case contradicts minimality. Hence every $a$ is a multiple of $b$, giving $I=(b)$. The zero ideal is $(0)$, so every ideal is a principal ideal and $R$ is a principal ideal domain.