# Diverging to Positive and Negative Infinity

Say I have some sequence $\{a_n\}$ with one subsequence $\{a_{n_i}\} \longrightarrow \infty$ and another $\{a_{n_j}\} \longrightarrow -\infty$. In other words, the lim sup $a_n = \infty$ and lim inf $a_n = -\infty.$

Because the sequence clearly does not converge, I am guessing I can call $\{a_n\}$ divergent. However, does $\{a_n\}$ diverge to $\infty$ and $-\infty$, or does it diverge to neither?

Just trying to make some sense of the definition of "divergence to infinity." My guess is that $\{a_n\}$ diverges, but does not diverge to either positive or negative infinity, since we can always find some element of the sequence greater than an arbitrary $M$ and another element less than $M$.

Many thanks.

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Consider the sequence $(-1)^n n$, it satisfies the conditions you mentioned at the beginning Is that what you want? –  Amr Nov 8 '12 at 1:01
Yes! That sequence seems to fit my description. Any ideas regarding the divergence issue? –  James Evans Nov 8 '12 at 1:02
The last part is true as well and the sequence I just gave satisfies the last part –  Amr Nov 8 '12 at 1:04
It diverges, full stop. Diverges to $\infty$ is just verbal shorhand for a particular kind of divergence behavior that a sequence of the kind that you’re talking about does not exhibit, and the same for diverges to $-\infty$. –  Brian M. Scott Nov 8 '12 at 1:05
So to say "diverges to infinity" does the following equality need to hold: lim inf = lim sup = $\infty$? –  James Evans Nov 8 '12 at 1:08

It is an abuse of terminology to say that the sequence "diverges to $+\infty$" or "diverges to $-\infty$", though people use it frequently.
What typically is meant by diverging to $+ \infty$ is the following:
$$\text{For any M>0, there exists N \in \mathbb{N} such that for all n > N, we have x_n > M.}$$
Similarly, for diverging to $-\infty$.