lowerlimit

Lower limit, in mathematics, is most often a colloquial term for the limit inferior of a sequence or function, denoted lim inf. It is defined as the greatest lower bound of the set of subsequential limits, equivalently liminf a_n = lim_{n→∞} inf_{k≥n} a_k. This value exists in the extended real numbers and satisfies lim inf a_n ≤ lim sup a_n. If a_n converges to L, then liminf a_n = limsup a_n = L, so the limit exists.

For monotone sequences, the lim inf equals the eventual limit: if a_n is nondecreasing, liminf a_n = lim a_n; if nonincreasing, liminf a_n = lim a_n. For functions, liminf f(x) as x→a is defined similarly via the infimum over punctured neighborhoods; if the limit exists, it equals the common value.

Examples: a_n = (-1)^n has liminf = -1 and limsup = 1; a_n = 1/n has liminf = limsup = 0. The concept is useful in real analysis, probability, and integration, and is closely related to the limsup (limit superior) and to lower bounds (infima).

In topology, the term lower limit may refer to the lower limit topology on the real line, also called the Sorgenfrey line. This topology is generated by half-open intervals [a,b) and has properties that differ from the standard topology, including being strictly finer and not second-countable.

Some older texts use lower limit to mean the infimum of a set, the greatest lower bound, though modern usage distinguishes between infimum and liminf. See also: lim inf, lim sup, lower bound, infimum, Sorgenfrey line.