Mengenlimits

Mengenlimits is a term used in mathematics to describe concepts related to the bounds and limiting behavior of sets. The basic idea is to quantify the extent to which a set fits within a range or to describe how a sequence of sets behaves in the limit.

In order theory and real analysis, a bound of a subset S of a partially ordered set is an element that is greater than or equal to all elements of S (upper bound) or less than or equal to all elements (lower bound). If S is nonempty and bounded above, it has a least upper bound, called the supremum; if it is nonempty and bounded below, it has a greatest lower bound, called the infimum. In the real numbers, sup A and inf A may be finite or infinite; for example, A = {1/n : n ∈ N} has infimum 0 and supremum 1.

For sequences of sets A_n, one defines limit superior and limit inferior: lim inf A_n = ∪_{n≥1} ∩_{k≥n} A_k and lim sup A_n = ∩_{n≥1} ∪_{k≥n} A_k. If these two sets coincide, the sequence converges to that limit. Other notions of set convergence exist in topology, notably the Kuratowski and Hausdorff limits, which describe convergence of sets in metric or topological spaces.

Examples and applications occur in optimization, probability theory, and measure theory, where one studies bounds, limits of random events, or convergence of measurable sets.

See also: supremum, infimum, bounds, lim inf, lim sup, Kuratowski convergence, Hausdorff convergence.