applicability: $\mathbb {R}$

Definition: Supremum, Least Upper Bound

Let \(D\) be a non-empty subset of real numbers. The real number \(\sup(D)\) is called the supremum (or the least upper bound) of \(D\), if

  1. \(\sup(D)\) is an upper bound of \(D\) and
  2. it is less than or equal to every upper bound of \(D\), i.e. if $B$ is any upper bound of $D$ then $\sup(D)\le B$.

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

Notes

Examples

Definitions: 1 2 3
Lemmas: 4
Proofs: 5 6 7 8 9 10 11
Propositions: 12 13
Theorems: 14


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References

Bibliography

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