Linfimum

Linfimum is a term encountered in some mathematical texts to denote a greatest lower bound associated with a linearly indexed family, typically interpreted as the limit inferior (liminf) of a sequence or net. Although liminf is the standard term in modern analysis, linfimum appears in older literature or in contexts that emphasize linear indexing of the index set.

For a real sequence (x_n) indexed by natural numbers, the linfimum is the liminf, defined as liminf x_n = sup_{n} inf_{k≥n} x_k. More generally, for a family {f_i} of real-valued functions on a common domain, indexed by a linearly ordered set I, the linfimum of {f_i} at a point x is liminf_i f_i(x) = sup_{i0∈I} inf_{i≥i0} f_i(x). If the indexed family is a convergent sequence, the linfimum equals the actual limit.

Key properties include that the linfimum is always a lower bound for the tail values of the sequence or net and equals the limit when the sequence converges. It is also less than or equal to any eventual lower bound of the sequence, and it provides information about the long-run lower behavior of the indexed collection.

In practice, linfimum is largely superseded by the standard terminology liminf. When encountered, it is helpful to recognize it as describing the same concept—the greatest lower bound approached by tail values—especially in contexts with linear indexing. See also infimum, liminf, limit superior, convergence.