Upper bound

An upper bound of a set in an ordered space is an element greater than or equal to every member of that set. It need not belong to the set or be its least upper bound.

Partial order: A relation that is reflexive, antisymmetric, and transitive. Upper bounds are defined relative to an order, which need not compare every pair of elements.
Least upper bound: An upper bound that is less than or equal to every other upper bound. It is the smallest element satisfying the upper-bound condition.
Completeness of the real numbers: The property that every nonempty real set bounded above has a least upper bound in the real numbers. This completeness property guarantees that bounded real sets have suprema.
Lower bound: An element less than or equal to every member of a specified set. It is the order-dual of an upper bound, reversing every comparison.
Set: A collection of distinct objects treated as a mathematical object. An upper bound is defined by checking every member of a specified set.
Supremum: The least upper bound of a set in an ordered structure. The supremum is the distinguished upper bound when a least one exists.
Bolzano–Weierstrass theorem: The theorem that every bounded sequence in Euclidean space has a convergent subsequence. Bounds on sequences enable a compactness result through convergent subsequences.
Infimum: The greatest lower bound of a set in an ordered structure. It is the lower-bound counterpart of a supremum.
Order relation: A binary relation expressing comparisons such as less than or equal to. The relation supplies the comparisons required in the definition of an upper bound.
Maximum element: An element of a set that is at least as large as every other element. A maximum is both a member of the set and an upper bound for it.

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