Quasi-increasing sequence that isn't monotone? How about one that diverges as well?

Question

A sequence ($a_n$) is called quasi-increasing if for all $\epsilon > 0$ there exists an $N$ such that whenever $n > m ≥ N$ it follows that $a_n > a_m − \epsilon$.

• Give an example of a sequence that is quasi-increasing but not monotone or eventually monotone.
• Give an example of a quasi-increasing sequence that is divergent and not monotone.
• Is there an analogue of the Monotone Convergence Theorem for quasi-increasing sequences? In other words, is it true that if a sequence is bounded and quasi-increasing then it must be convergent? Prove or give a counterexample.

Answer 1

HINTS:

• Show that every convergent sequence is quasi-increasing; once you have that, it’s easy to find an example. You want your sequence to oscillate.

• Modify your first example so that $\lim_na_n=\infty$, but the sequence still oscillates enough not to be monotone.

• It’s good strategy to try to prove it. If you succeed, you’re done, and if not, the failure of the attempt is likely to give you some insight into how to construct a counterexample. If you get completely stuck, there’s a bit more help in the spoiler-protected block below.

Yes, there is an analogue of the Monotone Convergence Theorem. Try to show that if $\langle a_n:n\in\Bbb N\rangle$ is quasi-increasing and bounded, then it converges to $\limsup_na_n$.