Does Every Bounded Divergent Sequence Oscillate? I'm struggling to reconcile my intuition with a rigorous proof. Taking an oscillating sequence $(s_n)$ to be one for which $\liminf_n(s_n)<\limsup_n(s_n)$, is it true that a bounded divergent sequence must oscillate? My intuition is that yes, it is.
 A: If the sequence is bounded, its elements lie in a (compact) interval $[a, b].$ Therefore it has limit points in the intervals. If all the limit points are the same, the sequence is convergent, if not, one is smaller than the other, so $\lim \inf < \lim \sup.$
A: We always have $\liminf(s_n) \le \limsup(s_n)$ and, in ${\Bbb R}^* = {\Bbb R} \cup \{ \infty, -\infty \}$, a sequence is convergent iff $\liminf(s_n) = \limsup(s_n)$. ${\Bbb R}$ being a subspace of ${\Bbb R}^*$, a sequence is convergent in ${\Bbb R}$ iff it is convergent in ${\Bbb R}^*$ and its limit lies in ${\Bbb R}$, i.e. the limit is finite. So a sequence is divergent (in ${\Bbb R}$) iff either it is divergent in ${\Bbb R}^*$ (i.e. $\liminf(s_n) < \limsup(s_n)$) or it is convergent in ${\Bbb R}^*$ but the limit does not lie in ${\Bbb R}$, i.e. the limit is infinite. Since a bounded sequence can't have an infinite limit, a bounded sequence is divergent (in ${\Bbb R}$) iff it is divergent in ${\Bbb R}^*$ i.e. iff $\liminf(s_n) < \limsup(s_n)$. So, in the terminology of your question, a bounded sequence is divergent iff it oscillates.
