Limit Inferior/Examples/Sequence of Reciprocals

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Example of Limit Inferior

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

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


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

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


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 inferior:

$\displaystyle \map {\liminf_{n \mathop \to \infty} } {a_n} = \inf \set 0 = 0$

$\blacksquare$


Sources