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Limit inferior (lim inf)

The smallest limit point of a bounded sequence, or equivalently the supremum of infima of tails.

The limit inferior (or lim inf) of a bounded sequence (xn)(x_n) in R\mathbb{R} is

lim inf⁡n→∞xn=lim⁡n→∞inf⁡k≥nxk=sup⁡n≥1inf⁡k≥nxk.\liminf_{n \to \infty} x_n = \lim_{n \to \infty} \inf_{k \geq n} x_k = \sup_{n \geq 1} \inf_{k \geq n} x_k.
Characterizations

For a (xn)(x_n), the lim inf equals:

  1. The smallest subsequential limit (when the sequence is bounded; this includes limits of eventually constant subsequences).
  2. The infimum of the set of all limits.
Properties
  • lim inf⁡xn≤lim sup⁡xn\liminf x_n \leq \limsup x_n always.
  • The sequence converges if and only if lim inf⁡xn=lim sup⁡xn\liminf x_n = \limsup x_n.
  • lim inf⁡(−xn)=−lim sup⁡(xn)\liminf(-x_n) = -\limsup(x_n).
  • lim inf⁡(xn+yn)≥lim inf⁡xn+lim inf⁡yn\liminf(x_n + y_n) \geq \liminf x_n + \liminf y_n (superadditivity).
Extended values

For an unbounded sequence, the lim inf is determined by the tail behavior: it is −∞-\infty when every tail is unbounded below. A finite initial outlier does not affect the lim inf. If xn→+∞x_n\to +\infty, then lim inf⁡xn=+∞\liminf x_n=+\infty.

Examples

For xn=(−1)n+1/nx_n = (-1)^n + 1/n: lim inf⁡xn=−1\liminf x_n = -1, lim sup⁡xn=1\limsup x_n = 1.