Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 1, Problem 4

Problem

Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $D$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.

Answer

Since $E$ is nonempty, there exists $x \in E$. Then $\alpha \leq x \leq \beta$.

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