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 Axiom of Archimedes:

$\exists N \in \N: N > H$

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.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (3) \ \text{(iii)}$