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:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$


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:

$\displaystyle \map {\lim_{n \mathop \to \infty} } {a_{n_r} } = 1$
$\displaystyle \map {\lim_{n \mathop \to \infty} } {a_{n_s} } = -1$


Hence by definition of limit superior:

$\displaystyle \map {\limsup_{n \mathop \to \infty} } {a_n} = 1$

$\blacksquare$


Sources