upperlimit

Upperlimit, often written as two words, generally refers to an upper bound in mathematics, though the term is used in several related ways. Broadly, an upper bound of a set S is any number u that is greater than or equal to every element of S. If S is bounded above, the least such bound is called the least upper bound or the supremum of S. In the real numbers, every nonempty set bounded above has a supremum, by the completeness property.

A related concept is the upper limit in the sense of limits of sequences. The limit superior, commonly abbreviated as lim sup, describes the largest accumulation value that a sequence can approach. Formally, for a sequence (a_n), lim sup a_n is the limit of the sequence of suprema of its tails: lim_{n→∞} sup_{k≥n} a_k, when this limit exists. It equivalently equals the supremum of the set of subsequential limits of (a_n). The lim inf (lower limit) plays a complementary role.

Examples help distinguish the concepts. For the set S = [0, 1), any upper bound is ≥ 1, and the supremum of S is 1. For the sequence a_n = (-1)^n, lim sup a_n = 1 and lim inf a_n = -1, illustrating how lim sup can differ from the actual limit.

In calculus, "upper limit of integration" refers to the upper endpoint of a definite integral. In programming or data contexts, upperlimit may appear as a variable name, but standard mathematical usage prefers "upper bound" or "supremum" depending on the context. See also lower bound, supremum, and lim sup.