Knowlpedia

Limit inferior

The eventual lower limiting value of a real sequence.

A limit inferior of a real sequence (an)n≥1(a_n)_{n\ge 1} is the extended real number

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,

provided the right-hand side is interpreted in [−∞,∞][-\infty,\infty].

Examples
  • If an=(−1)na_n=(-1)^n, then lim inf⁡n→∞an=−1\liminf_{n\to\infty} a_n=-1.
  • If an=1na_n=\tfrac1n, then lim inf⁡n→∞an=0\liminf_{n\to\infty} a_n=0.
Remarks

This definition is built from repeated use of and on the tails of the sequence. Together with the , it describes the full range of subsequential behavior via .