The Limit Superior
Table of Contents
1 Definition
We define the limit superior of a sequence \( a_n \) as follows:
\[ \limsup_{n \to \infty}{a_n} = \lim_{n \to \infty}{\left( \sup_{m \geq n}{a_m} \right)} \]
An equivalent but perhaps more intuitive definition is:
\[ \limsup_{n \to \infty}{a_n} = \sup{S} \]
Where \( S \) is the set of limits of all convergent subsequences of \( a_n \). The limit inferior is defined similarly:
\[ \liminf_{n \to \infty}{a_n} = \lim_{n \to \infty}{\left( \inf_{m \geq n}{a_m} \right)} \]
2 Interpretation
Consider some sequence \( a_n \) where the limit superior is not infinite. Then the limit superior \( b \) is the smallest real number s.t. \( \forall \epsilon > 0 \), there exist only a finite \( n \) s.t. \( a_n > b + \epsilon \).
3 Examples
Define \( a_n = \sin{n\pi} \)
4 Questions
Show that a sequence \( a_n \) converges iff:
\[ \limsup_{n \to \infty}{a_n} = \liminf_{n \to \infty}{a_n} \]
Show that if the sequences \( a_n \) and \( b_n \) are bounded above then:
\[ \limsup_{n \to \infty}{\left(a_n + b_n\right)} \le \limsup_{n \to \infty}{a_n} + \limsup_{n \to \infty}{b_n} \]
and find a similar relation for the limit inferior.