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Limits Supremum and Limits Infimum 📂Analysis

Limits Supremum and Limits Infimum

Definition

Let {xn}n∈N\left\{ x_{n} \right\}_{n \in \mathbb{N}}, {yn}n∈N\left\{ y_{n} \right\}_{n \in \mathbb{N}} be sequences of real numbers.

  1. lim sup⁡n→∞xn:=lim⁡n→∞(sup⁡k≥nxk)\displaystyle \limsup_{n \to \infty} x_{n} := \lim_{n \to \infty} \left( \sup_{k \ge n} x_{k} \right) is called the limit supremum of {xn}\left\{ x_{n} \right\}.
  2. lim inf⁡n→∞yn:=lim⁡n→∞(inf⁡k≥nyk)\displaystyle \liminf_{n \to \infty} y_{n} := \lim_{n \to \infty} \left( \inf_{k \ge n} y_{k} \right) is called the limit infimum of {yn}\left\{ y_{n} \right\}.

Where sup⁡k≥nxk:=sup⁡{xk:k≥n}\displaystyle \sup_{k \ge n} x_{k} := \sup \left\{ x_{k} : k \ge n \right\} and inf⁡k≥nxk:=inf⁡{xk:k≥n}\displaystyle \inf_{k \ge n} x_{k} := \inf \left\{ x_{k} : k \ge n \right\}.

Properties

  • (a): lim⁡k→∞xnk=x  ⟹  lim inf⁡n→∞xn≤x≤lim sup⁡n→∞xn \lim_{k \to \infty} x_{n_{k}} = x \implies \liminf_{n \to \infty} x_{n} \le x \le \limsup_{n \to \infty} x_{n}
  • (b): lim inf⁡n→∞xn=x=lim sup⁡n→∞xn  ⟺  lim⁡n→∞xn=x \liminf_{n \to \infty} x_{n} = x = \limsup_{n \to \infty} x_{n} \iff \lim_{n \to \infty} x_{n} = x
  • (c): −lim inf⁡n→∞xn=lim sup⁡n→∞(−xn)−lim sup⁡n→∞xn=lim inf⁡n→∞(−xn) \begin{align*} - \liminf_{n \to \infty} x_{n} =& \limsup_{n \to \infty} ( - x_{n} ) \\ - \limsup_{n \to \infty} x_{n} =& \liminf_{n \to \infty} ( - x_{n} ) \end{align*}
  • (d): If xn≤ynx_{n} \le y_{n}, then lim sup⁡n→∞xn≤lim sup⁡n→∞ynlim inf⁡n→∞xn≤lim inf⁡n→∞yn \begin{align*} \limsup_{n \to \infty} x_{n} \le& \limsup_{n \to \infty} y_{n} \\ \liminf_{n \to \infty} x_{n} \le& \liminf_{n \to \infty} y_{n} \end{align*}

Explanation

The concept of limit supremum is widely useful throughout analysis, introduced for convenience whether one agrees with it immediately or not. Intuitively, it helps to think about supremum and infimum while ‘discarding the initial part of the sequence’.

For example, when xk=1k\displaystyle x_{k} = {{ 1 } \over { k }}, let’s inspect the actual process of calculating sup⁡k≥n{xk}\displaystyle \sup_{k \ge n} \left\{ x_{k} \right\}.

n=3:sup⁡{13,14,15,⋯ }=13 n=3 : \sup \left\{ {{ 1 } \over { 3 }} , {{ 1 } \over { 4 }} , {{ 1 } \over { 5 }} , \cdots \right\} = {{ 1 } \over { 3 }}

n=4:sup⁡{14,15,⋯ }=14 n=4 : \sup \left\{ \quad {{ 1 } \over { 4 }} , {{ 1 } \over { 5 }} , \cdots \right\} = {{ 1 } \over { 4 }}

n=5:sup⁡{15,⋯ }=15 n=5 : \sup \left\{ \quad \quad {{ 1 } \over { 5 }} , \cdots \right\} = {{ 1 } \over { 5 }}

n→∞:sup⁡k≥n{1k:k∈N}=0 n \to \infty : \sup_{k \ge n} \left\{ {{ 1 } \over { k }} : k \in \mathbb{N} \right\} = 0

Thus, discarding the initial part implies that we are interested in speaking about a sufficiently large nn, and ultimately, it helps us understand its relevance to lim⁡\lim.

Given lim⁡n→∞{1k:k≥n}=∅\displaystyle \lim_{n \to \infty} \left\{ {{1} \over {k}} : k \ge n \right\} = \emptyset, therefore, not having sup⁡lim⁡n→∞{xn}\displaystyle \sup \lim_{n \to \infty} \left\{ x_{n} \right\} might make one appreciate why it’s sometimes necessary to introduce expressions like lim sup⁡n→∞=lim⁡n→∞sup⁡k≥n\displaystyle \limsup_{n \to \infty} = \lim_{n \to \infty} \sup_{k \ge n}. While limits are considered, it’s sufficient to have nn, hence it’s reasonable to think in terms of the limit of another sequence sns_{n}.

On the other hand, considering yn=1(−2)n−1\displaystyle y_{n} = {{1} \over {(-2)^{n-1}}} gives us sup⁡{yn}=1\sup \left\{ y_{n} \right\} = 1 and inf⁡{yn}=−12\displaystyle \inf \left\{ y_{n} \right\} = - {{1} \over {2}} but

lim sup⁡n→∞yn=lim inf⁡n→∞yn=0 \limsup_{n \to \infty} y_{n} = \liminf_{n \to \infty} y_{n} = 0

This serves as an example for property (b).