---
tag: 0B5H2V4JW8
kind: definition
name: Set bounded above
msc: [06A06]
defines: 0F0J193T76
sources:
  - url: https://en.wikipedia.org/wiki/Upper_and_lower_bounds
    locator: lead section
---

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $M$ of $P$ is an upper bound of $S$ when $s \le M$ for each $s$ in $S$.
The set $S$ is bounded above when it has at least one upper bound in $P$.
Similarly, $m$ is a lower bound of $S$ when $m \le s$ for each $s$ in $S$.
The set $S$ is bounded below when it has a lower bound.

## Remarks

In $\mathbb{R}$, the interval $[0, 1)$ is bounded above. Each number $M \ge 1$ is an upper bound.
The set $\mathbb{N}$ is not bounded above in $\mathbb{R}$, by the [[Archimedean property|0A74SX55QP]].
An upper bound of $S$ does not need to be an element of $S$.

## Explanations

### L1 human

A set is bounded above when it does not go up forever: some number is a ceiling for all its elements.
The set $\{1, 1/2, 1/3, \dots\}$ is bounded above by $1$, and also by $2$ and by $100$.