#math/analysis If $S$ is a nonempty subset of $\mathbb{R}$ and is bounded above, then in $\mathbb{R}$ there exists a `least upper bound` denoted $\sup S\in \mathbb{R}$. $\sup S$, also known as the `supremum`, may or may not belong to $S$. Proof: #TODO ## Supremum Inclusivity Test To determine whether $(a_{i})<\sup S$ or $(a_{i})\leq{}\sup S$. 1. Find $b=\sup S$. 2. Check whether $b\in S$. 1. If $b\in S$ then $b$ is also the `maximum` of $S$. 2. If $b\not\in S$, then $S$ has a supremum but no maximum. For example, $S=[0,1]\implies \sup S=1\in S\implies \max S=1,$ whereas, $ S=[0,1)\implies\sup S = 1\not\in S\implies \text{no maximum.} $