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.
Best Answer
We say a sequence diverges if it doesn't converge.
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 $-\infty$.