I cannot seem to think of any example of a convergent sequence that has divergent sub-sequences.
I am aware that a divergent sequence can have convergent sub-sequences. $(-1)^n$ is divergent and has convergent sub-sequences $(1,1,1,1...)$ or $(-1,-1,-1...)$
Can you please give me an example of the opposite case? Or is this simply not possible? If it not possible, how do I prove this?
