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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
- \(\sup(D)\) is an upper bound of \(D\) and
- 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
- Not all sets have the property that their non-empty subsets have a supremum. In fact, this definition is valid only for complete ordered fields, since in them, the existence of a supremum is ensured.
- For instance, the set of all rational numbers $0 \le x \le \sqrt{2}$ does not have a supremum since we cannot identify the least rational upper bound that is still $\le \sqrt{2},$ however, the set of all real numbers $0 \le x \le \sqrt{2}$ has the supremum $\sqrt{2},$ since it is the least real upper bound $\le \sqrt{2}.$
Examples
- The real number $b$ in the semi-open interval $[a,b)$ is the supremum of this interval because for every $\epsilon > 0$ (no matter how small) there is an $y\in[a,b)$ lying between the numbers $b-\epsilon$ and $b.$
- $\sqrt{2}\not\in\mathbb Q$ is the supremum of the set or real numbers whose square is less or equal $2$, i.e. the set $A:=\{x\in\mathbb Q\mid x^2\le 2\}$, because for every $\epsilon > 0$ there is a real number $d\in A$ with $d > \sqrt{2}-\epsilon.$
- Please note that the $\sup(D)$ does not have to be an element of $D$, like $b\not\in[a,b).$ However, $\sup(D)$ must be in the same domain as $D$, like $\sqrt{2}$ is not a supremum of $D=\{x\in\mathbb Q\mid x < \sqrt{2}\},$ but it is a supremum of $D=\{x\in\mathbb R\mid x < \sqrt{2}\}.$
Mentioned in:
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
- Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition