Limit inferior and limit superior

Limit Inferior and Limit Superior

In mathematics, particularly in the field of real analysis, the concepts of limit inferior and limit superior are used to describe the limiting behavior of sequences and functions. These concepts are essential in understanding the convergence properties of sequences and are widely used in various branches of mathematics.

Definitions

For a given sequence \( \{a_n\} \) of real numbers, the limit inferior and limit superior are defined as follows:

• The limit inferior of the sequence, denoted \( \liminf_{n \to \infty} a_n \), is the greatest lower bound (infimum) of the set of subsequential limits of \( \{a_n\} \).
• The limit superior of the sequence, denoted \( \limsup_{n \to \infty} a_n \), is the least upper bound (supremum) of the set of subsequential limits of \( \{a_n\} \).

These definitions can be expressed in terms of the sequence itself:

\[ \liminf_{n \to \infty} a_n = \lim_{n \to \infty} \left( \inf_{k \geq n} a_k \right) \]

\[ \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \left( \sup_{k \geq n} a_k \right) \]

These expressions highlight that the limit inferior is the limit of the infimum of the tail of the sequence, while the limit superior is the limit of the supremum of the tail of the sequence.

Properties

The limit inferior and limit superior have several important properties:

• \( \liminf_{n \to \infty} a_n \leq \limsup_{n \to \infty} a_n \) for any sequence \( \{a_n\} \).
• If \( \liminf_{n \to \infty} a_n = \limsup_{n \to \infty} a_n \), then the sequence \( \{a_n\} \) converges, and the common value is the limit of the sequence.
• The limit superior and limit inferior are both invariant under taking subsequences.

Examples

Consider the sequence \( \{(-1)^n\} \), which alternates between -1 and 1. The subsequential limits are -1 and 1. Therefore, the limit inferior is -1, and the limit superior is 1.

Example of limit superior and limit inferior

For the sequence \( \{1, 0, 1, 0, \ldots\} \), the limit inferior is 0, and the limit superior is 1, as the sequence oscillates between these two values.

Applications

Limit inferior and limit superior are used in various mathematical contexts, including:

• Analyzing the convergence of series and sequences in real analysis.
• Studying the behavior of functions in measure theory and probability theory.
• Investigating the stability of solutions in dynamical systems.

Visual Representation

The concepts of limit inferior and limit superior can be visualized using graphical representations. The following image illustrates these concepts:

Illustration of limit superior and limit inferior

Related Pages

• Convergence (mathematics)
• Real analysis
• Supremum and infimum
• Subsequence