Knowlpedia

Limit superior

The eventual upper limiting value of a real sequence.

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

lim sup⁡n→∞an  =  inf⁡n≥1 sup⁡k≥nak,\limsup_{n\to\infty} a_n \;=\; \inf_{n\ge 1}\,\sup_{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 sup⁡n→∞an=1\limsup_{n\to\infty} a_n=1.
  • If an=1na_n=\tfrac1n, then lim sup⁡n→∞an=0\limsup_{n\to\infty} a_n=0.
Remarks

This definition is built from repeated use of and on the “tails” of the sequence. It packages subsequential behavior: values near lim sup⁡an\limsup a_n are realized along suitable .