Infima

Infima

• The infimum of the set of numbers ${\displaystyle \{2,3,4\}}$ is ${\displaystyle 2.}$ The number ${\displaystyle 1}$ is a lower bound, but not the greatest lower bound, and hence not the infimum.
• More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set.
• ${\displaystyle \inf\{1,2,3,\ldots \}=1.}$
• ${\displaystyle \inf\{x\in \mathbb {R} :0<x<1\}=0.}$
• ${\displaystyle \inf \left\{x\in \mathbb {Q} :x^{3}>2\right\}={\sqrt[{3}]{2}}.}$
• ${\displaystyle \inf \left\{(-1)^{n}+{\tfrac {1}{n}}:n=1,2,3,\ldots \right\}=-1.}$
• If ${\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}$ is a decreasing sequence with limit ${\displaystyle x,}$ then ${\displaystyle \inf x_{n}=x.}$