# Limit Superior/Examples/Sequence of Reciprocals

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## 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:

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

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.14$: Example $\text {(i)}$