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

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.

## closed as off-topic by Frits Veerman, N. S., Adam Hughes, Cameron Williams, jgonOct 4 '16 at 21:38

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Frits Veerman, N. S., Adam Hughes, Cameron Williams, jgon
If this question can be reworded to fit the rules in the help center, please edit the question.

• For the curious, this is Exercise $\mathbf{2.6.6}$ in Stephen Abbott, Understanding Analysis. – Brian M. Scott Oct 4 '16 at 11:16

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