## Supremum (Least Upper Bound) The supremum of a set $S \subseteq \mathbb{R}$ is the smallest value that is greater than or equal to every element in $S$. More formally, a number $\sup S$ is called the supremum of $S$ if: 1. $\sup S \geq s$ for all $s \in S$ (i.e., it is an upper bound of $S$), and 2. for any $\epsilon > 0$, there exists some $s \in S$ such that $s > \sup S - \epsilon$ (i.e., it is the least upper bound). If $S$ has a maximum, then $\sup S$ is simply the maximum of $S$. If $S$ is unbounded above, it has no supremum. ## Infimum (Greatest Lower Bound) The infimum of a set $S \subseteq \mathbb{R}$ is the largest value that is less than or equal to every element in $S$. Formally, a number $\inf S$ is the infimum of $S$ if: 1. $\inf S \leq s$ for all $s \in S$ (i.e., it is a lower bound of $S$), and 2. for any $\epsilon > 0$, there exists some $s \in S$ such that $s < \inf S + \epsilon$ (i.e., it is the greatest lower bound). If $S$ has a minimum, then $\inf S$ is simply the minimum of $S$. If $S$ is unbounded below, it has no infimum.