Infimum

analysis set-theory

Definition

Infimum

The infimum (or greatest lower bound) of a set $S\subseteq \mathbb{R}$ is the largest real number that is less than or equal to every element in $S$. It is denoted by $\inf S$.

Formally, $L=\inf S$ if:

1. $L$ is a lower bound: $\forall x\in S:x\ge L$.
2. $L$ is the greatest such bound: if $M$ is any lower bound of $S$, then $L\ge M$.

If the set $S$ is not bounded below, we define $\inf S=-\infty$. For the empty set, $\inf \varnothing =\infty$.

Relation to Minimum

An infimum is called a minimum if it is an element of the set $S$. While every non-empty set that is bounded below has an infimum (see Completeness of the Real Numbers), it does not necessarily have a minimum (e.g., the open interval $(0,1)$ has an infimum of $0$ but no minimum).

Properties

• Uniqueness: If an infimum exists, it is unique.
• Approximation Property: For any $\epsilon >0$, there exists an element $x\in S$ such that $x<(\inf S)+\epsilon$.
• Sequences: The infimum of a sequence $(a_n)$ is the infimum of the set of its values $\{a_n\mid n\in \mathbb{N}\}$.