> [!NOTE] **Definition** (Supremum & Infimum) >Given a set of [[Real numbers|reals]] $S \subset \mathbb{R}$, $l$ is the *least upper bound* (or *supremum*) of $S$ iff: >1. $l$ an upper bound for $S$ and >2. if $L$ is any upper bound of $S$ then $l \leq L$. > > We say $l$ is the *greatest upper bound* (or *Infimum*) of $S$ iff >1. $l$ a lower bound for $S$ and >2. if $L$ is any upper bound of $S$ then $l \leq L$. ^de37d4 # Properties By [[Characteristic Property of Supremum of Subset of Real Numbers]]