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The function g is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set. The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a point d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper bound for f.
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral (). of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit.
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bound s, that is, = {: ({: <}) =} if the set of essential lower bounds is nonempty, and as otherwise; again there is an alternative expression as = {: ()} (with this being if the set is empty).