# Limit Superior/Examples/Sequence of Reciprocals

Jump to navigation Jump to search

## Example of Limit Superior

Let $\sequence {a_n}$ be the sequence defined as:

$\forall n \in \N_{>0}: a_n = \dfrac 1 n$

The limit superior of $\sequence {a_n}$ is given by:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = 0$

## Proof

From Sequence of Reciprocals is Null Sequence, $\sequence {a_n}$ is convergent:

$\displaystyle \lim_{n \mathop \to \infty} \dfrac 1 n = 0$

Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {a_n}$.

By Limit of Subsequence equals Limit of Real Sequence, all such subsequences have limit $0$.

Hence by definition of limit superior:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = \sup \set 0 = 0$

$\blacksquare$