Well-ordered sets have the least upper bound property

Let \(A\) be a well-ordered set, and let \(B\subset A\) be nonempty and bounded above. The set \[ U=\{u\in A:b\le u\text{ for every }b\in B\} \] of upper bounds is nonempty. Since \(A\) is well-ordered, \(U\) has a smallest element \(u_0\). By definition \(u_0\) is an upper bound of \(B\), and no smaller element is an upper bound. Hence \[ u_0=\sup B. \]