Supremum

definitionreal-analysis

Definition

We say that $M\in \mathbb{R}$ is the supremum of a set $S\subset \mathbb{R}$ if the following are true,

1. $M$ is an upper bound of $S$ (therefore $S$ is bounded above).
2. For any upper bound $M^{\prime }$ of $S$, $M\le M^{\prime }$ ($M$ is unique).

$M$ is also called the least upper bound.

We denote the supremum of a set $S$, $\sup (S)$.

Examples