Assume we have a sequence $ a_n $ such that any one of $ \lim inf $ or $ \lim sup $ exist. Let’s say $ \lim inf =l $ exist( this can be $\lim sup$ also .Then we know that at least one subsequence converges to $ l$ . But what I am proposing is that every subsequence either converges or tends to infinity .
No subsequence can oscillate randomly and be bounded . Since if even one such subsequence exist then $ \lim inf $ or $ \lim sup $ cannot be uniquely determined .
Am I right ?