Upper and lower bounds

[4] For example, 5 is a lower bound for the set S = {5, 8, 42, 34, 13934} (as a subset of the integers or of the real numbers, etc.Every finite subset of a non-empty totally ordered set has both upper and lower bounds.Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f(x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x.The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.An upper bound u of a subset S of a preordered set (K, ≤) is said to be an exact upper bound for S if every element of K that is strictly majorized by u is also majorized by some element of S. Exact upper bounds of reduced products of linear orders play an important role in PCF theory.
A set with upper bounds and its least upper bound
Big O notationmathematicsorder theorysubsetpreordered setgreater than or equal toDuallyintegersreal numbersnatural numbersrational numberstotally ordered setfunctionsdomaincodomainsupremuminfimumreduced productslinear ordersPCF theoryGreatest element and least elementInfimum and supremumMaximal and minimal elementsSchaefer, Helmut H.Mac Lane, SaundersBirkhoff, GarrettAmerican Mathematical Society