liminfn

Liminf n, commonly written as liminfn, is an informal shorthand for the limit inferior of a sequence indexed by n. In formal terms, for a real sequence (a_n), liminf a_n is defined as lim_{n→∞} inf_{k≥n} a_k, which is also equal to sup_{n} inf_{k≥n} a_k. The standard notation in mathematical texts is liminf_{n→∞} a_n.

Intuition and basic meaning: The liminf captures the eventual lower bound of a sequence. If you look at the tails of the sequence and take the smallest value each tail attains, the limit of those tail infima gives the liminf. Equivalently, it is the greatest lower bound among all subsequential limit points of (a_n).

Properties: The sequence b_n = inf_{k≥n} a_k is nondecreasing, so the limit lim_{n→∞} b_n exists in the extended real numbers. The liminf is finite when (a_n) is bounded below; otherwise it is −∞. It satisfies liminf a_n ≤ limsup a_n, where limsup a_n is the limit of the corresponding tail suprema.

Relation to subsequences: The set of subsequential limits of (a_n) has infimum equal to liminf a_n and supremum equal to limsup a_n. Thus liminf is the greatest lower accumulation point (in a precise sense) of the sequence, and it bounds all possible subsequential limits from below.

Examples: If a_n = 1/n, then liminf a_n = 0 and limsup a_n = 0. If a_n = (−1)^n, then liminf a_n = −1 and limsup a_n = 1. If a_n = −n, then liminf a_n = −∞ and limsup a_n = ∞.

Notes: While liminfn is a convenient shorthand in informal discussion, the formal notation remains liminf_{n→∞} a_n. It is a fundamental concept in real analysis and probability theory.