Limit Superior

analysis

Definition

Limit Superior

The limit superior (or upper limit) of a sequence $(a_n{)}_{n\ge 0}$ is the largest limit point of the sequence, including improper limit points. It is denoted by:

$$\underset{n\to \infty }{\mathrm{limsup}}{a}_n=\underset{n\ge 0}{\inf }\underset{k\ge n}{\sup }{a}_k$$

Every sequence of real numbers has a limit superior in the extended real number system $\mathbb{R}\cup \{-\infty ,\infty \}$.

Properties

• Relation to Limit: A sequence converges to $L$ if and only if $\mathrm{liminf}{a}_n=\mathrm{limsup}{a}_n=L$.
• Boundedness: For any $\epsilon >0$,$a_n<(\mathrm{limsup}{a}_n)+\epsilon$ for almost all $n$, and $a_n>(\mathrm{limsup}{a}_n)-\epsilon$ for infinitely many $n$.
• Inequality: It always holds that ${\mathrm{liminf}}_{n\to \infty }{a}_n\le {\mathrm{limsup}}_{n\to \infty }{a}_n$.