Supremum

Last updated Nov 1, 2022

Let $(T \leq)$ be a Total Ordering. Let $A \subset T$. Set $U = {x \in T : x \geq a \text{ } \forall a \in A}$. That is, $U$ is the Set of Upper Bounds of $A$. Then $\sup A$ (if it exists) is the Minimum of $U$.

1. If $A \subset T$ is s.t. $A = \emptyset$, then examining the definition of $U$, we see that the condition holds vacuously and $U = T$. Thus $\sup \emptyset = \min T$ if the minmum exists.
2. Let $A \subset \mathbb{R}$. For some $u \in \mathbb{R}$, write $u \geq A$ if $\forall a \in A$, $u \geq a$. One of the equivalent conditions for Completeness of the Real Numbers is that every $A \subset \mathbb{R}$ s.t. $\exists u \in \mathbb{R}$ with $u \geq A$ has a Supremum in $\mathbb{R}$.

• Sum of Supremum bounds Supremum of Sum
• There is a Monotone Net Converging to Supremum
• Non-Decreasing Continuous Functions preserve Supremum
• Non-Increasing Continuous Functions send Supremum to Infimum
• Negative Multiplication sends Supremum to Infimum
• Non-negative Multiplication Commutes with Supremum
• Supremum of Negative is Negative of Infimum

• Real Numbers
• Empty Set