Let be a Euclidean Domain. Let be minimal among the norms of nonzero elements of ( exists because every nonempty set of natural numbers has a least element). Prove that every nonzero element of norm is a unit. Deduce that a nonzero element of norm zero (if it exists) is a unit.
We denote the Euclidean norm on by . Choose with and . By the Division Algorithm, we have for some , with . Since is minimal among the norms of nonzero elements in , we have . Thus , so that is a unit.
Suppose now that some nonzero element has norm zero. Then zero is necessarily minimal among the norms of nonzero elements in , and hence any nonzero element of norm zero is a unit.