Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence is denoted by and the limit superior of a sequence is denoted by The of a sequence (xn) is defined by or Similarly, the of (xn) is defined by or Alternatively, the notations and are sometimes used. The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence . An element of the extended real numbers is a subsequential limit of if there exists a strictly increasing sequence of natural numbers such that . If is the set of all subsequential limits of , then and If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice. Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist.