Limit Superior/Examples/(-1)^n

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