---
tag: 04KFTA34C5
kind: definition
name: Infimum
msc: [06A06]
defines: 0QF1MZ9FRG
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 $a$ of $P$ is an infimum, or greatest lower bound, of $S$ when these two conditions hold:

1. $a$ is a [[lower bound|0B5H2V4JW8]] of $S$;
2. if $m$ is a lower bound of $S$, then $m \le a$.

The notation is $a = \inf S$.

## Remarks

A set has at most one infimum.
In $\mathbb{R}$, $\inf \{1/n : n \ge 1\} = 0$, and $0$ is not an element of this set.
For a subset $S$ of $\mathbb{R}$, $\inf S = -\sup(-S)$ when one side exists, where $-S = \{-s : s \in S\}$.

## Explanations

### L1 human

The infimum is the highest possible floor for a set.
For the set $\{1, 1/2, 1/3, \dots\}$, every number at most $0$ is a floor, and $0$ is the highest of these, so the infimum is $0$.