Limsup

Limsup, short for limit superior, is a concept associated with real or extended real sequences {a_n}. It can be viewed as the largest limit point of the sequence or as the limit of the sequence of tail suprema. Formally, limsup_{n→∞} a_n denotes lim_{n→∞} sup_{k≥n} a_k. Since sup_{k≥n} a_k is nonincreasing in n, this limit exists in the extended real line, taking values in [-∞, +∞].

Equivalently, limsup a_n is the supremum of the set of subsequential limits: limsup a_n = sup{L : there exists a subsequence a_{n_j} → L} when convergence is interpreted in the extended real numbers. If all subsequences diverge to +∞, the limsup is +∞; if they diverge to -∞, the limsup is -∞.

Basic properties include that liminf a_n ≤ limsup a_n, and the sequence has a (finite) limit if and only if these two quantities are equal. For any constants c, limsup(a_n + c) = limsup a_n + c, and limsup(c a_n) = c limsup a_n when c ≥ 0. In general, limsup satisfies the inequality limsup(a_n + b_n) ≤ limsup a_n + limsup b_n.

Examples help illustrate: for a_n = 0 when n is even and 1 when n is odd, limsup a_n = 1 and liminf a_n = 0. For a_n = n, limsup a_n = +∞ (and likewise liminf a_n = +∞). For a_n = (-1)^n, limsup a_n = 1 and liminf a_n = -1.

Extensions include defining limsup pointwise for sequences of functions and studying limsup in probability, where it describes almost-sure convergence properties of random variables.