Limit Superior/Examples/(-1)^n (1 + n^-1)

Example of Limit Superior
Let $\sequence {a_n}$ be the sequence defined as:
 * $\forall n \in \N_{>0}: a_n = \paren {-1}^n \paren {1 + \dfrac 1 n}$

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


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

This is not the same as:


 * $\displaystyle \sup_{n \mathop \ge 1} \paren {-1}^n \paren {1 + \dfrac 1 n}$

Proof
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {a_n}$.

We have that:

and:


 * $\sequence {a_n}$ contains no subsequence which converges to a limit different from $-1$ or $1$.

Hence:
 * $L = \set {-1, 1}$

and it follows that:


 * $\displaystyle \map {\limsup_{n \mathop \to \infty} } {\paren {-1}^n \paren {1 + \dfrac 1 n} } = 1$

But note that:


 * $\displaystyle \sup_{n \mathop \ge 1} \paren {-1}^n \paren {1 + \dfrac 1 n}$

occurs when $n = 2$, at which point:


 * $\paren {-1}^n \paren {1 + \dfrac 1 n} = \paren {-1}^2 \paren {1 + \dfrac 1 2} = \dfrac 3 2$