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Definition: Infimum, Greatest Lower Bound

Let \(D\) be a non-empty subset of real numbers. The real number \(\inf(D)\) is called the infimum (or the greatest lower bound) of \(D\), if

1. \(\inf(D)\) is a lower bound of \(D\) and
2. it is greater than or equal to every lower bound of \(D\), i.e. if $B$ is any lower bound of $D$ then \(\inf(D) \ge B\).

Equivalently, for every \(\epsilon > 0\) there exists an \(y\in D\) with \(\inf(D) + \epsilon > y\).

Example

• The real number $a$ in the semi-open interval $(a,b]$ is the infimum of this interval because for every $\epsilon > 0$ (no matter how small) there is an $y\in(a,b]$ lying between the numbers $a$ and $a+\epsilon.$
• Please note that the $\inf(D)$ does not have to be an element of $D$, like $a\not\in(a,b].$

Mentioned in:

Definitions: 1 2
Proofs: 3 4 5 6 7
Propositions: 8 9

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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition