limit superior
or upper limit (abbrev. lim sup), n. 1. the limit as n tends to infinity of the suprema of the subsequences of the elements beyond the nth of a given real sequence:
lim sup n→ ∞ an = limn→ ∞ [sup {am : m ≥ n}].
This produces the largest cluster point of the sequence, and may be positive infinity. The sequence has a limit if and only if the limit superior and the limit inferior are identical, and in that case the limit is their common value. 2. the set of points that are in infinitely many members of a given sequence of sets {An}
lim sup n→ ∞ an = limn→ ∞ [sup {am : m ≥ n}].
This produces the largest cluster point of the sequence, and may be positive infinity. The sequence has a limit if and only if the limit superior and the limit inferior are identical, and in that case the limit is their common value. 2. the set of points that are in infinitely many members of a given sequence of sets {An}
lim sup n→ ∞ An = ∩m {∪n≥m An}, m=1...∞.
The sets are said to have a limit if the limit superior and limit inferior agree, and the limit is then this common set. There are topological analogues of these concepts.