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varliminf

Denotes the limit inferior (greatest lower bound of limit points) of a sequence or function.

Overview

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

  • Particularly important in real analysis and topology for characterizing oscillating sequences.
  • Provides a way to analyze the most conservative limiting behavior of a sequence.
  • Commonly used alongside supremum and infimum operations in convergence proofs.
  • Appears frequently in measure theory and functional analysis when studying sequence properties.

Examples

Definition of liminf for a sequence of real numbers.

lim‾⁡n→∞an=sup⁡n≥1inf⁡k≥nak\varliminf_{n \to \infty} a_n = \sup_{n \geq 1} \inf_{k \geq n} a_k 
\varliminf_{n \to \infty} a_n = \sup_{n \geq 1} \inf_{k \geq n} a_k

Comparing liminf and limsup of a sequence.

lim‾⁡n→∞xn≤lim sup⁡n→∞xn\varliminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n 
\varliminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n

Using varliminf in a convergence inequality.

lim‾⁡n→∞(an+bn)≥lim‾⁡n→∞an+lim‾⁡n→∞bn\varliminf_{n \to \infty} (a_n + b_n) \geq \varliminf_{n \to \infty} a_n + \varliminf_{n \to \infty} b_n 
\varliminf_{n \to \infty} (a_n + b_n) \geq \varliminf_{n \to \infty} a_n + \varliminf_{n \to \infty} b_n