Liminf

Liminf, or limit inferior, is a concept in real analysis that describes the greatest lower bound of the tail values of a sequence, or the greatest lower bound of function values near a point. For a sequence (a_n), it is defined by liminf_{n→∞} a_n = sup_{n≥1} inf_{k≥n} a_k. The inner infimum takes the minimum of the tail from index n onward, and the outer supremum collects these tail-lower-bounds as n grows. The definition extends to the extended real line, allowing liminf to be -∞ or ∞ when appropriate.

For a real-valued function f defined near a real point a, the limit inferior as x approaches a is liminf_{x→a} f(x) = sup_{δ>0} inf { f(x) : 0<|x−a|<δ }. This formulation captures the greatest lower bound of f on shrinking punctured neighborhoods of a.

Liminf is paired with the limit superior (limsup), defined by limsup_{n→∞} a_n = lim_{n→∞} sup_{k≥n} a_k = inf_{n} sup_{k≥n} a_k. In general, liminf a_n ≤ limsup a_n, and the two coincide if and only if the sequence converges to a finite limit (in which case both equal that limit).

Examples help illustrate: for a_n = (-1)^n, liminf a_n = -1 and limsup a_n = 1. For a_n = 1/n, both liminf and limsup equal 0. If a_n = (-1)^n n, liminf = -∞ and limsup = ∞. Liminf also constrains subsequential limits: any limit of a subsequence lies between liminf a_n and limsup a_n.