TeXipedia

varlimsup

Denotes the limit superior (supremum limit) of a sequence, representing the largest possible limit point of a sequence.

Overview

Essential in advanced mathematical analysis for describing the behavior of sequences and series where traditional limits may not exist.

  • Commonly used in real analysis to characterize sequence convergence properties
  • Appears frequently in measure theory and probability theory
  • Particularly useful when dealing with sequences that oscillate or don't converge in the traditional sense
  • Often paired with its counterpart varliminf when analyzing sequence boundaries

Examples

Upper limit superior of a sequence.

lim‾⁡n→∞an=L\varlimsup_{n \to \infty} a_n = L 
\varlimsup_{n \to \infty} a_n = L

Showing the relationship between limsup and regular limit.

lim⁡n→∞xn=L  ⟺  lim‾⁡n→∞xn=lim‾⁡n→∞xn=L\lim_{n \to \infty} x_n = L \iff \varlimsup_{n \to \infty} x_n = \varliminf_{n \to \infty} x_n = L 
\lim_{n \to \infty} x_n = L \iff \varlimsup_{n \to \infty} x_n = \varliminf_{n \to \infty} x_n = L

Definition of limsup using infimum of supremums.

lim‾⁡n→∞xn=inf⁡n≥1sup⁡k≥nxk\varlimsup_{n \to \infty} x_n = \inf_{n \geq 1} \sup_{k \geq n} x_k 
\varlimsup_{n \to \infty} x_n = \inf_{n \geq 1} \sup_{k \geq n} x_k