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Limit superior (lim sup)

The largest limit point of a bounded sequence, or equivalently the infimum of suprema of tails.

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

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

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

  1. The largest subsequential limit (when the sequence is bounded; this includes limits of eventually constant subsequences).
  2. The supremum 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, and then the limit equals this common value.
  • lim sup⁡(−xn)=−lim inf⁡(xn)\limsup(-x_n) = -\liminf(x_n).
Extended values

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

Examples

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