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.