Divergent Real Sequence to Infinity/Examples/(-1)^n times n

Example of Divergent Real Sequence to Infinity
Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:


 * $a_n = \paren {-1} n$

Then $\sequence {a_n}$ is divergent to $\infty$.

However, $\sequence {a_n}$ is neither divergent to $+\infty$ nor divergent to $-\infty$.

Proof
We have that:
 * $\size {a_n} = n$

Let $H \in \R_{>0}$ be given.

Then by the Archimedean Principle:
 * $\exists N \in \N: N > H$

and so for $n \ge N$:


 * $n > H$

Thus $\sequence {a_n}$ is divergent to $\infty$.

But $\sequence {a_n}$ cannot be divergent to $+\infty$ because all its odd terms are negative.

Neither can $\sequence {a_n}$ cannot be divergent to $-\infty$ because all its even terms are positive.

Hence the result.