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:
- $\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 0$
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 $\text {(i)}$