Consider a bounded sequence whose terms are eventually "outgrown" by other terms of the sequence, i.e., for any index there exists an such that . Does this property imply that is eventually monotonic, i.e., that there exists an index such that is monotonically increasing from index onwards? I can't think of any counterexample. If true, do you know if this a known theorem which I can reference from some standard textbook?
NB: the boundedness assumption is essential, otherwise you can construct simple counterexamples like