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.