Ülempiir

Ülempiir is a term used in Estonian mathematics to denote a quantity that serves as an upper bound for a set of numbers. Specifically, given a set S of real numbers, a number M is an upper bound of S if every element x in S satisfies x ≤ M. The collection of all upper bounds of S exists whenever S is bounded above. The least upper bound, if it exists, is called the supremum of S, denoted sup S, and it is the smallest among all upper bounds.

In the real numbers, by the completeness axiom, every nonempty set that is bounded above has a supremum. The supremum may belong to S (in which case S has a maximum) or it may lie strictly above all elements of S.

Examples help illustrate the idea. Let S = {1, 3, 4}. Then 4 is an upper bound and also the supremum, so sup S = 4. If S = (0, 1), then 1 is an upper bound and the supremum, even though 1 is not an element of S.

Uses and significance: Ülempiir is central in analysis and topology, underpinning concepts of limits, convergence, integration, and optimization. It clarifies how sets can be contained within bounds from above and how limits behave with respect to these bounds.

See also: upper bound, least upper bound, supremum, bound.

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