---
tag: 0BTMQ04556
kind: definition
name: Supremum
msc: [06A06]
defines: 0R3H5PANRQ
uses: [0B5H2V4JW8]
sources:
  - url: https://en.wikipedia.org/wiki/Infimum_and_supremum
    locator: Formal definition
---

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $b$ of $P$ is a supremum, or least upper bound, of $S$ when these two conditions hold:

1. $b$ is an [[upper bound|0B5H2V4JW8]] of $S$;
2. if $M$ is an upper bound of $S$, then $b \le M$.

The notation is $b = \sup S$.

## Remarks

A set has at most one supremum, see [[uniqueness of the supremum|04ETX30HT2]].
In $\mathbb{R}$, $\sup [0, 1) = 1$ and $\sup [0, 1] = 1$. The first set does not contain its supremum.
In $\mathbb{Q}$, the set $\{q \in \mathbb{Q} : q^2 < 2\}$ is bounded above but has no supremum.

## Explanations

### L1 human

The supremum is the lowest possible ceiling for a set.
For the set $\{0.9, 0.99, 0.999, \dots\}$, every number at least $1$ is a ceiling, and $1$ is the lowest of these, so the supremum is $1$.