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Limit inferior

The supremum of the infima of the tails of an extended-real sequence.

The limit inferior of a sequence (an)n≥1(a_n)_{n\ge1} of extended is

lim inf⁡n→∞an  =  sup⁡n≥1 inf⁡k≥nak,\liminf_{n\to\infty} a_n \;=\; \sup_{n\ge 1}\,\inf_{k\ge n} a_k,

where the supremum and infima are taken in [−∞,+∞][-\infty,+\infty].

Remarks

The tail infima form a nondecreasing sequence, so their extended-real supremum is always defined. Together with the , the limit inferior describes asymptotic subsequential behavior.