Let \(E\) be a nonempty subset of an ordered set; suppose \(\alpha\) is a lower bound of \(E\) and \(\beta\) is an upper bond of \(E\). Prove that \(\alpha\le\beta\).
\(E\) is nonempty, thus there exists some element, \(x \in E\). Because \(\alpha\) is a lower bound of \(E\), \(\alpha \le x\). Because \(\beta\) is an upper bound of \(E\), \( x \le \beta\).
By transitivity of the ordering relation, \(\alpha \le x\) and \(x \le \beta\) imply that \(\alpha \le \beta \).