---
tag: 0H5J02P8S9
kind: definition
name: Upper bound
msc: [06A06]
defines: 0S60DVMVPN
uses: [011C29Q1S1]
sources:
  - url: https://en.wikipedia.org/wiki/Upper_and_lower_bounds
    locator: lead section
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $u \in P$ is an upper bound of $S$ when $s \le u$ for each $s \in S$.
The set $S$ is bounded above when it has an upper bound.

## Remarks

An upper bound need not be in $S$. The number $1$ is an upper bound of $[0, 1)$ in $\mathbb{R}$, and so is $5$.
A lower bound is defined in the same way with $u \le s$.
The least upper bound of $S$, when it exists, is the supremum of $S$.

## Explanations

### L1 human

An upper bound is a ceiling for the set: no element of the set is above it.
The set $\{1, 1/2, 1/3, \dots\}$ has upper bounds $1$, $2$, $100$ and many more.