Limit Superior/Examples/(-1)^n
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Example of Limit Superior
Let $\sequence {a_n}$ be the sequence defined as:
- $\forall n \in \N_{>0}: a_n = \paren {-1}^n$
The limit superior of $\sequence {a_n}$ is given by:
- $\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$
Proof
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {a_n}$.
From Divergent Sequence may be Bounded, $\sequence {a_n}$ is bounded above by $1$ and bounded below by $-1$.
We have the subsequences:
- $(1): \quad \sequence {a_{n_r} }$ where $\sequence {n_r}$ is the integer sequence defined as $n_r = 2 r$
- $(2): \quad \sequence {a_{n_s} }$ where $\sequence {n_s}$ is the integer sequence defined as $n_s = 2 s + 1$.
We have that:
- $\sequence {a_{n_r} }$ is the sequence $1, 1, 1, 1, \ldots$
- $\sequence {a_{n_s} }$ is the sequence $-1, -1, -1, -1, \ldots$
and so:
- $\ds \map {\lim_{n \mathop \to \infty} } {a_{n_r} } = 1$
- $\ds \map {\lim_{n \mathop \to \infty} } {a_{n_s} } = -1$
Hence by definition of limit superior:
- $\ds \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Lim sup and lim inf: $\S 5.14$: Example $\text {(ii)}$