limit inferior
or lower limit (abbrev. lim inf), n. 1. the limit as n tends to infinity of the infima of the subsequences of the elements beyond the nth of a given real sequence:
lim infn→ ∞ an = limn→ ∞ [inf {am : m ≥ n}].
This produces the smallest cluster point of the sequence, and may be negative infinity. The sequence has a limit if and only if the limit inferior and limit superior are identical, and in that case the limit is their common value. 2. the set of points that are in all but finitely many members of a given sequence of sets {An}:
lim infn→ ∞ An = ∪m{∩n≥m An}, m=1...∞.
The sets are said to have a limit if the limit inferior and limit superior agree, and the limit is then this common set. For example, the limit of a nested decreasing sequence of sets is the intersection of the collection. There are topological analogues of these concepts.
lim infn→ ∞ an = limn→ ∞ [inf {am : m ≥ n}].
This produces the smallest cluster point of the sequence, and may be negative infinity. The sequence has a limit if and only if the limit inferior and limit superior are identical, and in that case the limit is their common value. 2. the set of points that are in all but finitely many members of a given sequence of sets {An}:
lim infn→ ∞ An = ∪m{∩n≥m An}, m=1...∞.
The sets are said to have a limit if the limit inferior and limit superior agree, and the limit is then this common set. For example, the limit of a nested decreasing sequence of sets is the intersection of the collection. There are topological analogues of these concepts.