Limit Superior/Examples/Sequence of Reciprocals

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:

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

Proof

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

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

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:

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

$\blacksquare$

Sources

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